" Mansfield and Wildberger say that if their interpretation is correct, Plimpton 322 would not only be the oldest known trigonometric table, but it would also be the "world's only completely accurate trigonometric table.
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"You don't make a trigonometric table by accident," Dr. Mansfield said.
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Further evidence that the Babylonians put the trigonometric principles supposedly contained in the table to use in their society or another tablet depicting the same trigonometric table would be needed to put Mansfield's interpretation beyond a doubt.
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The key to observing such a distant point was figuring out trigonometric parallaxes.
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This ancient trigonometric method measures the angles in a triangle formed by three survey control points.
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Conversely, if I know the value of one of these trigonometric functions, I can easily determine the value of the angle.
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I hope the sight of the six trig functions didn't trigger PTSD (Post Trigonometric Stress Disorder) in any solvers out there.
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Mansfield's interpretation of the Plimpton tablet is considered controversial and far from total proof that the tablet is indeed a trigonometric table.
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A British geographer making a trigonometric survey gave the mountain its name, a clipped abstraction capturing its indifference to life and time.
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The actual Brigsby Bear Adventures TV show is clearly rated G, in spite of some educational messages about penis-touching and trigonometric functions.
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As Manfield points out, this method is both more exact and simpler than the trigonometric system invented by the Greeks 1,000 years later.
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"It's a trigonometric table, which is 245,256 years ahead of its time," said Daniel F. Mansfield of the University of New South Wales.
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This Babylonian tablet, known as Plimpton 23, is the world's oldest—and most accurate—trigonometric table, according to research published this week in Historia Mathematica.
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But ironically enough the final presentation of Brown's tables had the same sum-of-trigonometric-functions form that one would get from having lots of epicycles.
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Once I was in lecture class and decided to take that day to learn about this girl instead of about the differentiation of inverse trigonometric functions.
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The ancient trigonometric method of triangulation to measure a mountain's height (where the angles in a triangle formed by three survey control points) will not be used.
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Without further evidence, many scholars concluded that the tablet was not a trigonometric table used for determining the ratios of a triangle's sides, but an ancient school text.
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Last year, for example, scientists revealed the secrets of a mysterious 3,700-year-old Babylonian clay tablet, which they describe as the world's oldest and most accurate trigonometric table.
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I zoomed in and out, trying to find a readable scale, but the path wheeled frantically on the screen as the device succumbed to a kind of trigonometric panic.
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Juventus looked the more adventurous side in the first 20 minutes but it was Real who struck first after the sort of trigonometric build-up that coaches write books about.
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In their paper, Dr. Mansfield and Dr. Wildberger show that this is better than what would be calculated using a trigonometric table from the Indian mathematician Madhava 3,000 years later.
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A teacher at Sydney's Cherrybrook Technology High School, Woo has more than 40,000 subscribers — not bad for someone whose content is dedicated to explaining concepts like trigonometric equations, inequality proofs, and checksums.
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As you've probably noticed in the above table, the values of the trigonometric functions are approximate—most of the ratios in the table will have values that extend far beyond six decimal places.
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But according to Daniel Mansfield, a mathematician at New South Wales and the lead author of the new paper, seeing the tablet as a trigonometric table just required a shifting of the frame of reference.
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The trigonometric functions used to describe these ratios are called sine, cosine, and tangent, which represent the ratio of the opposite side to the hypotenuse, adjacent side to the hypotenuse, and the opposite side to the adjacent side, respectively.
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The last time I cracked open a fortune cookie, my fortune said something weak about optimism and taught me the Chinese characters for zebra, which immediately passed all six trigonometric functions on the list of information I will never use.
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So instead of using the trigonometric functions we're familiar with today, they used the ratios of the short to long side and the diagonal side to the long side of the triangle to generate a wide variety of right angle triangles.
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Perhaps the strongest argument in favor of the hypothesis of Dr. Mansfield and Dr. Wildberger is that the table works for trigonometric calculations, that someone had put in the effort to generate Pythagorean triples to describe right triangles at roughly one-degree increments.
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But the problem turned out to be with the calculations, and by 1748 Euler was using sums of about 20 trigonometric terms and proudly proclaiming that the tables he'd produced for the three-body problem had predicted the time of a total solar eclipse to within minutes.
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An early entrant was Leonhard Euler, who developed methods based on trigonometric series (including much of our current notation for such things), and whose works contain many immediately recognizable formulas: In the mid-203s there was a brief flap—also involving Euler's "competitors" Clairaut and d'Alembert—about the possibility that the inverse-square law for gravity might be wrong.
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His "Taxi Cab" series, riots of impasto geometry, were an unequivocal "feh" to the fashionable ideas of the New York School and the first inkling of what would become his signature hard-lined, "concrete" style — something that caught favor with people like Frank Stella and Ellsworth Kelly, and which Held eventually developed into rigorous, near-trigonometric excursions into space and perspective.
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The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.
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The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.
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An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
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In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions.
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Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments (see trigonometric function).
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In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods. This form is especially suited for interpolation of periodic functions.
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In transcendental models, the output variable is computed by solving transcendental equations, namely equations involving trigonometric, inverse trigonometric, exponential, logarithmic, and/or hyperbolic functions.
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Si(x) (blue) and Ci(x) (green) plotted on the same plot. In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions.
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In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
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In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e^{ix} and e^{-ix} and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts, and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
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In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
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This implies that a trigonometric number is an algebraic number, and twice a trigonometric number is an algebraic integer. Ivan Niven gave proofs of theorems regarding these numbers.Niven, Ivan. Numbers: Rational and Irrational, 1961.
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These trigonometric functions can themselves be expanded, using multiple angle formulae.
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The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the 30-60-90 triangle which can be used to evaluate the trigonometric functions for any multiple of π/6, and the 45-45-90 triangle which can be used to evaluate the trigonometric functions for any multiple of π/4.
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Trigonometric calculations played an important role in the early study of astronomy. Early tables were constructed by repeatedly applying trigonometric identities (like the half-angle and angle-sum identities) to compute new values from old ones.
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Existence and uniqueness for trigonometric interpolation now follows immediately from the corresponding results for polynomial interpolation. For more information on formulation of trigonometric interpolating polynomials in the complex plane, see p. 135 of Interpolation using Fourier Polynomials.
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Bari's last work—her 55th publication—was a 900-page monograph on the state of the art of trigonometric series theory, which is recognized as a standard reference work for those specializing in function and trigonometric series theory.
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If and are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for and can be obtained from expansion of these functions in the appropriate trigonometric series.
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Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).
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It hosted a trigonometric station during the Swedish-Russian Arc-of-Meridian Expedition.
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In mathematics, the Denjoy–Luzin theorem, introduced independently by and states that if a trigonometric series converges absolutely on a set of positive measure, then the sum of its coefficients converges absolutely, and in particular the trigonometric series converges absolutely everywhere.
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Chebfun, a fully integrated software system written in MATLAB for computing with functions, uses trigonometric interpolation and Fourier expansions for computing with periodic functions. Many algorithms related to trigonometric interpolation are readily available in Chebfun; several examples are available here.
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Another proof is based on differential equations satisfied by exponential and trigonometric functions. See .
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Trigonometric sine and cosine functions used 1.4 degree precision (256 values) via look-up tables.
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It is evident that a trigonometric function has the same value for all coterminal angles.
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Positivstellensatz also exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.
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They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis. The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.
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The most complete and influential trigonometric work of antiquity is the Almagest of Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416. Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
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Geometry Revisited, pages 60-63. a trigonometric one, a symmetry-based approach, and proofs using complex numbers.
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The trigonometric parallax of WISE 0359−5401 is arcsec, corresponding to a direct inversion distance of , or .
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Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.
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For the 25 years preceding the invention of the logarithm in 1614, prosthaphaeresis was the only known generally applicable way of approximating products quickly. It used the identities for the trigonometric functions of sums and differences of angles in terms of the products of trigonometric functions of those angles.
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Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.
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In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions.
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Cosines and sines around the unit circle In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified.
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During the Meroitic period in Nubian history the ancient Nubians used a trigonometric methodology similar to the Egyptians.
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All types of electronic targets use some form of trigonometric equations to triangulate the position of bullet impact.
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This equivalence explains why linear combinations are called polynomials. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.
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The first trigonometric table was apparently compiled by Hipparchus, who is consequently now known as "the father of trigonometry".
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A point on the mountain was used as a trigonometric point during the Swedish- Russian Arc-of-Meridian Expedition.
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This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics.
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In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (), the versine () (which appeared in the earliest tables), the coversine (), the haversine (), the exsecant (), and the excosecant (). See List of trigonometric identities for more relations between these functions.
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Trigonometric parallax of WISE 0535−7500 is 0.070 ± 0.005 arcsec, corresponding to a distance of 14 pc and 47 ly.
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In mathematics, Ulugh Beg wrote accurate trigonometric tables of sine and tangent values correct to at least eight decimal places.
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Bowring showed that the single iteration produces a sufficiently accurate solution. He used extra trigonometric functions in his original formulation.
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Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.
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Why the ubiquity of the "bell-shaped curve"? There is a theoretical reason for this, and it involves Fourier transforms and hence trigonometric functions. That is one of a variety of applications of Fourier transforms to statistics. Trigonometric functions are also applied when statisticians study seasonal periodicities, which are often represented by Fourier series.
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Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled- triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval , see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity.
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In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in degrees (particularly in elementary mathematics). When using trigonometric function in calculus, their argument is generally not an angle, but a real number. In this case, it is more suitable to express the argument of the trigonometric as the length of the arc of the unit circle—delimited by an angle with the center of the circle as vertex.
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This led to an essentially new estimate of trigonometric sums and a new mean value theorem for such systems of equations.
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That is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values.
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Reflecting θ in α=0 (α=) By examining the unit circle, one can establish the following properties of the trigonometric functions.
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A college preparatory/regular class might focus on topics used in business-related careers, such as matrices, or power functions. A standard course considers functions, function composition, and inverse functions, often in connection with sets and real numbers. In particular, polynomials and rational functions are developed. Algebraic skills are exercised with trigonometric functions and trigonometric identities.
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Jyā, koti-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematicians and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine.
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The versine or versed sine is a trigonometric function found in some of the earliest (Vedic Aryabhatia I) trigonometric tables. The versine of an angle is 1 minus its cosine. There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation.
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They are used also in the discrete Fourier transform. The term trigonometric polynomial for the real-valued case can be seen as using the analogy: the functions sin(nx) and cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of eix.
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The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is :\sin^2 \theta + \cos^2 \theta = 1. As usual, means (\sin\theta)^2.
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Other commonly used trigonometric identities include the half- angle identities, the angle sum and difference identities, and the product-to- sum identities.
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Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant.
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274 in his De triangulis omnimodis written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed. The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. In the 17th century, Isaac Newton and James Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions. In the 18th century, Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "Euler's formula" eix = cos x + i sin x.
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One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
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The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astronomy, and spherical trigonometry and could help improve accuracy, but are rarely used today except to simplify some calculations. A unit circle with trigonometric functions.
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These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations to have much worse accuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some FFTs other than Cooley–Tukey, such as the Rader–Brenner algorithm, are intrinsically less stable.
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The front side is designed as a Gunter quadrant and the rear side as a trigonometric quadrant. The side with the astrolabe has hour lines, a calendar, zodiacs, star positions, astrolabe projections, and a vertical dial. The side with the geometric quadrants features several trigonometric functions, rules, a shadow quadrant, and the chorden line.Ralf Kern: Wissenschaftliche Instrumente in ihrer Zeit.
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Gyrotrigonometry is the use of gyroconcepts to study hyperbolic triangles. Hyperbolic trigonometry as usually studied uses the hyperbolic functions cosh, sinh etc., and this contrasts with spherical trigonometry which uses the Euclidean trigonometric functions cos, sin, but with spherical triangle identities instead of ordinary plane triangle identities. Gyrotrigonometry takes the approach of using the ordinary trigonometric functions but in conjunction with gyrotriangle identities.
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The oldest definitions of trigonometric functions, related to right- angle triangles, define them only for acute angles. To extending these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) is often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations.
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This is not immediately evident from the above geometrical definitions. Moreover, the modern trend in mathematics is to build geometry from calculus rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus. For defining trigonometric functions inside calculus, there are two equivalent possibilities, either using power series or differential equations.
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"Solution of triangles" is the main trigonometric problem: to find missing characteristics of a triangle (three angles, the lengths of the three sides etc.) when at least three of these characteristics are given. The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc.
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The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is :\sin^2 x + \cos^2 x = 1 .
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Trigonometria contains about 36 pages of writing. In this book, the abbreviations for the trigonometric functions are explained in further detail consisting of mathematical tables.
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The sixth chapter contains series expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and inverse tangent functions.
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In mathematics, a trigonometric number is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of , or the sine or cosine of a rational number of degrees. One of the simplest examples is \cos \frac \pi 4=\frac \sqrt 2 2. A real number different from is a trigonometric number if and only if it is the real part of a root of unity (see Niven's theorem). Thus every trigonometric number is half the sum of two complex conjugate roots of unity.
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A trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. If sin(nx) and cos(nx) are expanded in terms of sin(x) and cos(x), a trigonometric polynomial becomes a polynomial in the two variables sin(x) and cos(x) (using List of trigonometric identities#Multiple-angle formulae). Conversely, every polynomial in sin(x) and cos(x) may be converted, with Product-to-sum identities, into a linear combination of functions sin(nx) and cos(nx).
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The Great Arc refers to the systematic exploration and recording of the entire topography of the Indian subcontinent which was spearheaded by the Great Trigonometric Survey.
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The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.
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The formulae below involve finite sums; for infinite summations or finite summations of expressions involving trigonometric functions or other transcendental functions, see list of mathematical series.
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Fourth, this article only deals with trigonometric function values when the expression in radicals is in real radicals – roots of real numbers. Many other trigonometric function values are expressible in, for example, cube roots of complex numbers that cannot be rewritten in terms of roots of real numbers. For example, the trigonometric function values of any angle that is one-third of an angle θ considered in this article can be expressed in cube roots and square roots by using the cubic equation formula to solve :4\cos^3 \frac \theta 3 - 3\cos \frac \theta 3 = \cos\theta, but in general the solution for the cosine of the one-third angle involves the cube root of a complex number (giving casus irreducibilis). In practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Trigonometric tables.
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11, 1956. Cambridge University Press (2005): . Li Zhou and Lubomir Markov recently improved and simplified Niven's proofs. Any trigonometric number can be expressed in terms of radicals.
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Charlesfjellet is a mountain in Prins Karls Forland, Svalbard that is 969 m.a.s.l. tall and located on Grampianfjella. In 1910, it was used as a trigonometric station.
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A space cardioid introduction at Georgia Tech The space cardioid is a 3-dimensional curve derived from the cardioid. It has a parametric representation using trigonometric functions.
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The Haar system is a basis for . The trigonometric system is a basis in when . The Schauder system is a basis in the space .see p. 3.
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Long Range's primary military specialty is Thunderclap driver, and his secondary military specialty is artillery. As a kid, he struggled with simple arithmetic, but was able to master complex trigonometric and calculus problems with ease. He joined the Army's Artillery corps, where he utilized his trigonometric skills to amass the highest percentage rate of on-target knock outs the corps had ever seen and earn the nickname "The Knock Out Man".
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In the second method, he draws a real life object with a step-by-step process. In each step, he tries to find out which mathematical formulas will produce the drawing. For example, by using this method, he drew birds in flight, butterflies, human faces and plants using trigonometric functions. Naderi Yeganeh says: "In order to create such shapes, it is very useful to know the properties of the trigonometric functions".
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Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function.
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The book Divine Proportions shows the application of calculus using rational trigonometric functions, including three-dimensional volume calculations. It also deals with rational trigonometry's application to situations involving irrationals, such as the proof that Platonic Solids all have rational 'spreads' between their faces.See Divine Proportions for numerous examples of calculus done with rational trigonometric functions, as well as problems involving the application of rational trigonometry to situations containing irrationals.
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Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions using Padé approximant, and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series. Another possibility is to use wavelets. The Whittaker–Shannon interpolation formula can be used if the number of data points is infinite or if the function to be interpolated has compact support.
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WISE 0350−5658 is one of the nearest known brown dwarfs: its trigonometric parallax is 0.184 ± 0.010 arcsecond, corresponding to a direct distance of 5.4 pc (17.7 ly).
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The HP-35 was Hewlett-Packard's first pocket calculator and the world's first scientific pocket calculator: a calculator with trigonometric and exponential functions. It was introduced in 1972.
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"The Book of Origins: The first of everything – from art to zoos". Hachette UK which he developed to calculate logarithmic tangents.Eli Maor (2013). "Trigonometric Delights", Princeton University Press.
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It was powered by a National Semiconductor MM57134ENW/M integrated circuit. The President Scientific added logarithmic and trigonometric functions. It was powered by a General Instrument CF-599.
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In the above the "sign" function is +1 for positive arguments, -1 for negative arguments and zero at zero. It should not be confused with the trigonometric sine function.
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In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis.
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The first tables of trigonometric functions known to be made were by Hipparchus (c.190 - c.120 BCE) and Menelaus (c.70–140 CE), but both have been lost.
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Dmitrii Evgenevich Menshov (also spelled Men'shov, Menchoff, Menšov, Menchov; ; 18 April 1892 – 25 November 1988) was a Russian mathematician known for his contributions to the theory of trigonometric series.
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Consequently, :\sin^2 x + \cos^2 x = 1 \ , which is the Pythagorean trigonometric identity. When the trigonometric functions are defined in this way, the identity in combination with the Pythagorean theorem shows that these power series parameterize the unit circle, which we used in the previous section. This definition constructs the sine and cosine functions in a rigorous fashion and proves that they are differentiable, so that in fact it subsumes the previous two.
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Plot of the six trigonometric functions, the unit circle, and a line for the angle radians. The points labelled , , represent the length of the line segment from the origin to that point. , , and are the heights to the line starting from the -axis, while , , and are lengths along the -axis starting from the origin. The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions.
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It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set Sn of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S. If Sk+1 is the set of limit points of Sk, then he could construct a trigonometric series whose zeros are Sk+1. Because the sets Sk were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets S, S1, S2, S3,... formed a limit set, which we would now call Sω, and then he noticed that Sω would also have to have a set of limit points Sω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω, ω + 1, ω + 2, ... Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers.
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The successor of Lexell's trigonometric approach (as opposed to a coordinate approach) was Swiss mathematician L'Huilier. Both L'Huilier and Lexell emphasized the importance of polygonometry for theoretical and practical applications.
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Edward Schaumberg Quade (28 June 1908 – 4 June 1988)California, Death Index, 1940-1997 was an American mathematician at the Rand Corporation who worked on trigonometric series and systems analysis.
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Most mathematical functions commonly used by engineers, scientists and navigators, including logarithmic and trigonometric functions, can be approximated by polynomials, so a difference engine can compute many useful tables of numbers.
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Trigonometric numbers are irrational cosines or sines of angles that are rational multiples of . Such a number is constructible if and only if the denominator of the fully reduced multiple is a power of or the product of a power of with the product of one or more distinct Fermat primes. Thus, for example, Is constructible because is the product of two Fermat primes, and . See here a list of trigonometric numbers expressed in terms of square roots.
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Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy and celestial navigation. They continued to be widely used until electronic calculators became cheap and plentiful, in order to simplify and drastically speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks, and specialized tables were published for numerous applications.
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Luzin's first significant result was a construction of an almost everywhere divergent trigonometric series with monotonic convergence to zero coefficients (1912). This example disproved the Pierre Fatou conjecture and was unexpected to most mathematicians at that time. At approximately the same time, he proved what is now called Lusin's theorem in real analysis. His Ph.D. thesis titled Integral and trigonometric series (1915) had a large impact on the subsequent development of the metric theory of functions.
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Nina Karlovna Bari (, November 19, 1901, Moscow – July 15, 1961, Moscow) was a Soviet mathematician known for her work on trigonometric series.Biography of Nina Karlovna Bari, by Giota Soublis, Agnes Scott College.
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The first Ramsden theodolite was built for this survey. The survey was finally completed in 1853. The Great Trigonometric Survey of India began in 1801. The Indian survey had an enormous scientific impact.
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Distance of WISE J0521+1025 was estimated by Bihain et al. using mean absolute magnitudes of single T7.5 dwarfs, derived by Dupuy & Liu (2012) from trigonometric parallaxes: 5.0 ± 1.3 pc (16.3 ± 4.2 ly).
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Currently the most accurate distance estimate of WISE 2056+1459 is a trigonometric parallax, published in 2014 by Beichman et al.: 0.140 ± 0.009 arcsec, corresponding to a distance 7.1 pc, or 23.3 ly.
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Currently the most accurate distance estimate of WISE 0410+1502 is a trigonometric parallax, published in 2014 by Beichman et al.: 0.160 ± 0.009 arcsec, corresponding to a distance 6.3 pc, or 20.4 ly.
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Currently the most accurate distance estimate of WISE 2056+1459 is a trigonometric parallax, published in 2014 by Beichman et al.: 0.128 ± 0.010 arcsec, corresponding to a distance 7.8 pc, or 25.5 ly.
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The backsight level reading is always subtracted from the foresight level reading, and that equals the change in elevation between the two points, which is why this method is referred to as differential levelling. The other method of levelling in surveying is called trigonometric levelling, which involves the direct use of the total station (or any outdated instrument that reads angles), to read the vertical (zenith) angle to a prism rod, and uses the trigonometric functions to obtain the change in elevation.
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However, this involves costly inverse trigonometric functions, which generally makes this algorithm slower than the ray casting algorithm. Luckily, these inverse trigonometric functions do not need to be computed. Since the result, the sum of all angles, can add up to 0 or 2\pi (or multiples of 2\pi) only, it is sufficient to track through which quadrants the polygon winds,. as it turns around the test point, which makes the winding number algorithm comparable in speed to counting the boundary crossings.
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The list in this article is incomplete in several senses. First, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here. Second, it is always possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle, etc. Third, expressions in real radicals exist for a trigonometric function of a rational multiple of if and only if the denominator of the fully reduced rational multiple is a power of 2 by itself or the product of a power of 2 with the product of distinct Fermat primes, of which the known ones are 3, 5, 17, 257, and 65537.
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3); Vernet (2008) That Majorcan navigators had some sort of trigonometric table at hand is not improbable. Nonetheless, the exact content and layout of this table implied by Ramon Llull in 1295 is uncertain.
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The original 1984 price was 99 GBP or 199 CAD and included one Datapak and one software DATAPAK, the "Utility" pack. This latter adds scientific and trigonometric functions to the otherwise basic calculator routines.
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He remained in correspondence with Lichtenberg throughout his career. Klügel made an exceptional contribution to trigonometry, unifying formulae and introducing the concept of trigonometric function, (including coining the term) in his Analytische Trigonometrie 1770.
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These two limits are used in proofs of the fact that the derivative of the sine function is the cosine function. That fact is relied on in other proofs of derivatives of trigonometric functions.
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This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane (from which some isolated points are removed).
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The Cowriga Creek (technically a river) rises about southeast of Huntley trigonometric station west of Spring Hill, and flows generally south and south by west before reaching its confluence with the Belubula River northwest of .
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The game uses the DSP-1 chip, which is the same chip used by Super Mario Kart. The DSP chip provides fast support for the floating point and trigonometric calculations needed by 3D math algorithms.
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In this sections denote the three (interior) angles of a triangle, and denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
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The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and Fourier transforms.
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Supported common mathematical functions (unary, binary and variable number of arguments), including: trigonometric functions, inverse trigonometric functions, logarithm functions, exponential function, hyperbolic functions, Inverse hyperbolic functions, Bell numbers, Lucas numbers, Stirling numbers, prime-counting function, exponential integral function, logarithmic integral function, offset logarithmic integral , binomial coefficient and others. Expression e = new Expression("sin(0)+ln(2)+log(3,9)"); double v = e.calculate(); Expression e = new Expression("min(1,2,3,4)+gcd(1000,100,10)"); double v = e.calculate(); Expression e = new Expression("if(2<1, 3, 4)"); double v = e.
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Maor, Eli. Trigonometric Delights, Princeton Univ. Press, 2000 The Egyptians of those times apparently did not know the Pythagorean theorem; the only right triangle whose proportions they knew was the 3:4:5 triangle.Bell, Eric Temple.
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147, pp. 63–65, 1978. and others. Prior to the development of digital computers, lens optimization was a hand- calculation task using trigonometric and logarithmic tables to plot 2-D cuts through the multi-dimensional space.
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The trigonometric functions can be constructed geometrically in terms of a unit circle centered at O. Historically, the versed sine was considered one of the most important trigonometric functions. As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient. Even with a calculator or computer, round-off errors make it advisable to use the sin2 formula for small θ. Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle (θ = 0, 2, …) where it is zero--thus, one could use logarithmic tables for multiplications in formulas involving versines.
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Stars and brown dwarfs closest to the Sun, including WISE 1049−5319 (or Luhman 16), as of 2014. The trigonometric parallax of Luhman 16 as published by Sahlmann & Lazorenko (2015) is arcsec, corresponding to a distance of .
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In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful.
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Simon Sidon or Simon Szidon (1892 in Versec, Kingdom of Hungary – 27 April 1941, Budapest, Hungary) was a reclusive Hungarian mathematician who worked on trigonometric series and orthogonal systems and who introduced Sidon sequences and Sidon sets.
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Avoiding the use of expensive trigonometric functions improves speed over the basic form. It discards of the total input uniformly distributed random number pairs generated, i.e. discards uniformly distributed random number pairs per Gaussian random number pair generated, requiring input random numbers per output random number. The basic form requires two multiplications, 1/2 logarithm, 1/2 square root, and one trigonometric function for each normal variate.Note that the evaluation of 2U1 is counted as one multiplication because the value of 2 can be computed in advance and used repeatedly.
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Harmukh was first climbed by members of the Great Trigonometric Survey led by Thomas Montgomerie in 1856. Montgomerie made the first survey of the Karakoram some to the north, and sketched the two most prominent peaks, labeling them K1 and K2. The policy of the Great Trigonometric Survey was to use local names for mountains wherever possibleThe most obvious exception to this policy was Mount Everest, where the local name Chomolungma was probably known, but ignored in order to pay tribute to George Everest. See Curran, p. 29-30.
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A plot of f(x) = \sin(x) and g(x) = \cos(x); both functions are periodic with period 2π. The trigonometric functions sine and cosine are common periodic functions, with period 2π (see the figure on the right). The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.
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For example, the element labeled cb is a base program that implements the core functionality of a calculator. Element gb adds code that displays the core functionality as a GUI; element wb adds code that displays the core functionality via the web. Similarly, element ct adds trigonometric code to the core calculator functionality; elements gt and wt add code to display the trigonometric functionality as a GUI and web front-ends. A calculator is uniquely specified by two sequences of features: one sequence defines the calculator functionality, the other the front-end.
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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of the electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation sn for sin.
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Ordinary trigonometry studies triangles in the Euclidean plane R2. There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space.
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The phase-amplitude converter creates the sample-domain waveform from the truncated phase output word received from the PA. The PAC can be a simple read only memory containing 2M contiguous samples of the desired output waveform which typically is a sinusoid. Often though, various tricks are employed to reduce the amount of memory required. This include various trigonometric expansions, trigonometric approximations and methods which take advantage of the quadrature symmetry exhibited by sinusoids. Alternatively, the PAC may consist of random access memory which can be filled as desired to create an arbitrary waveform generator.
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Signs of trigonometric functions in each quadrant In the above graphic, the words in quotation marks are a mnemonic for remembering which three trigonometric functions (sine, cosine and tangent) are positive in each quadrant. The expression reads "All Science Teachers Crazy" and proceeding counterclockwise from the upper right quadrant, we see that "All" functions are positive in quadrant I, "Science" (for sine) is positive in quadrant II, "Teachers" (for tangent) is positive in quadrant III, and "Crazy" (for cosine) is positive in quadrant IV. There are several variants of this mnemonic.
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In the 1930s, he was a visiting scholar to lecture at the Jacques Hadamard's seminar at the Collège de France. His research touched real analysis, probability and mathematical statistics, in particular focused on the Fourier series. Rajchman received significant results in the fields of trigonometric series, function of a real variable and probability. In mathematics, there are such concepts as the Rajchman global uniqueness theorem, Rajchman measures, Rajchman collection, Rajchman algebras, Rajchman sharpened law of large numbers, Rajchman theory of formal multiplication of trigonometric series, Rajchman inequalities, and Rajchman- Zygmund inequalities.
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Rational trigonometry makes nearly all problems solvable with only addition, subtraction, multiplication or division, as trigonometric functions (of angle) are purposefully avoided in favour of trigonometric ratios in quadratic form. At most, therefore, results required as distance (or angle) can be approximated from an exact-valued rational equivalent of quadrance (or spread) after these simpler operations have been carried out. To make use of this advantage however, each problem must either be given, or set up, in terms of prior quadrances and spreads, which entails additional work.Olga Kosheleva (2008), "Rational trigonometry: computational viewpoint", Geombinatorics, Vol.
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Popov 2008 Some of the seven have very weak optical counterparts. For the brightest one (RX J1856-3754), the trigonometric parallax and proper motion are known.Kaplan et al. 2002 The distance to the sources is about 161 parsecs.
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Hipparchus, credited with compiling the first trigonometric table, has been described as "the father of trigonometry". Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.Aaboe, Asger (2001). Episodes from the Early History of Astronomy.
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Distortion is very low as well. It is not a standard projection in the sense that it uses complex polynomials (of the tenth order) rather than a trigonometric formulation, though it was developed from an oblique stereographic projection.
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Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identitiesSuch as the derivation of the formula for \tan (\alpha + \beta) from the addition formulas of sine and cosine. and hypergeometric identities.Petkovsek et al. 1996.
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The sixth chapter of Karanapaddhati is mathematically very interesting. It contains infinite series expressions for the constant π and infinite series expansions for the trigonometric functions. These series also appear in Tantrasangraha and their proofs are found in Yuktibhāṣā.
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The polar form requires 3/2 multiplications, 1/2 logarithm, 1/2 square root, and 1/2 division for each normal variate. The effect is to replace one multiplication and one trigonometric function with a single division and a conditional loop.
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Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge. The discovery did much to set him on the course for developing an abstract, general theory of infinite sets.
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Trigonometric parallax of this object, measured in 2001–2002 with the USNO 61 inch (1.5 m) reflector under US Naval Observatory (USNO) parallax program, is 0.1765 ± 0.0005 arcsec, corresponding to a distance of 5.67 ± 0.02 pc, or 18.48 ± 0.05 ly.
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M7 gun director 1944 A director, also called an auxiliary predictor,, Lone Sentry is a mechanical or electronic computer that continuously calculates trigonometric firing solutions for use against a moving target, and transmits targeting data to direct the weapon firing crew.
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The purpose of CTIOPI is to discover nearby red, white, and brown dwarfs that lurk unidentified in the solar neighborhood. The goal is to discover 300 new southern star systems within 25 parsecs by determining trigonometric parallaxes accurate to 3 milliarcseconds.
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Thus a single table of common logarithms can be used for the entire range of positive decimal numbers.E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). See common logarithm for details on the use of characteristics and mantissas.
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Find a grave - Simeon Borden Most, if not all, of the original field notes from Borden's trigonometric survey of Massachusetts are in the possession of the Massachusetts Association of Land Surveyors and Civil Engineers, Inc., One Walnut Street, Boston, MA.
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Sang worked for many years on trigonometric and logarithmic tables. Summaries of his tables were published by Alex Craik. Sang's 1871 table and his project for a 9-place million table were (re)constructed as part of the LOCOMAT project.
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The Global Ozone Monitoring by Occultation of Stars (GOMOS) is a measurement instrument aboard the European satellite Envisat. It measures ozone amounts by using the emitted electromagnetic spectrum from surrounding stars combined with trigonometric calculations in a process called stellar occultation.
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Victor Lenard Shapiro (16 October 1924, Chicago – 1 March 2013, Riverside, California) was an American mathematician, specializing in trigonometric series and differential equations. He is known for his two theorems (published in 1957) on the uniqueness of multiple Fourier series.
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Various types of equations can be solved using trigonometry. For example, a linear difference equation or linear differential equation with constant coefficients has solutions expressed in terms of the eigenvalues of its characteristic equation; if some of the eigenvalues are complex, the complex terms can be replaced by trigonometric functions of real terms, showing that the dynamic variable exhibits oscillations. Similarly, cubic equations with three real solutions have an algebraic solution that is unhelpful in that it contains cube roots of complex numbers; again an alternative solution exists in terms of trigonometric functions of real terms.
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The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function f(x) on the interval [0, 2\pi], which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero. Later Cantor proved that even if the set S on which f is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result.
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In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis.. They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the unit interval. But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions. The system of Walsh functions is known as the Walsh system.
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Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002) The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.
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Later, I. M. Vinogradov extended the technique, replacing the exponential sum formulation f(z) with a finite Fourier series, so that the relevant integral In is a Fourier coefficient. Vinogradov applied finite sums to Waring's problem in 1926, and the general trigonometric sum method became known as "the circle method of Hardy, Littlewood and Ramanujan, in the form of Vinogradov's trigonometric sums".Mardzhanishvili (1985), pp. 387–8 Essentially all this does is to discard the whole 'tail' of the generating function, allowing the business of r in the limiting operation to be set directly to the value 1.
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The primary solution angles in the form (cos,sin) on the unit circle are at multiples of 30 and 45 degrees. Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. All trigonometric numbers – sines or cosines of rational multiples of 360° – are algebraic numbers (solutions of polynomial equations with integer coefficients); moreover they may be expressed in terms of radicals of complex numbers; but not all of these are expressible in terms of real radicals. When they are, they are expressible more specifically in terms of square roots.
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This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial. Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order. This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is a further generalization where only derivatives of some orders are prescribed, not necessarily all orders from 0 to a k.
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Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric series. The first edition was a single volume, published in 1935 (under the slightly different title Trigonometrical Series). The second edition of 1959 was greatly expanded, taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is similar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square integrable functions.
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If all the operations are performed with high precision, the product can be as accurate as desired. Although sums, differences, and averages are easy to compute with high precision, even by hand, trigonometric functions and especially inverse trigonometric functions are not. For this reason, the accuracy of the method depends to a large extent on the accuracy and detail of the trigonometric tables used. For example, a sine table with an entry for each degree can be off by as much as 0.0087 if we just round an angle off to the nearest degree; each time we double the size of the table (for example by giving entries for every half-degree instead of every degree) we halve this error. Tables were painstakingly constructed for prosthaphaeresis with values for every second, or 3600th of a degree. Inverse sine and cosine functions are particularly troublesome, because they become steep near −1 and 1.
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G 240-72 is the seventh closest white dwarf (after Sirius B, Procyon B, van Maanen's star, Gliese 440, 40 Eridani B and Stein 2051 B). Its trigonometric parallax is 0.1647 ± 0.0024 arcsec, corresponding to a distance 6.07 ± 0.09 pc, or 19.80 ly.
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Traian Lalescu (; 12 July 1882 – 15 June 1929) was a Romanian mathematician. His main focus was on integral equations and he contributed to work in the areas of functional equations, trigonometric series, mathematical physics, geometry, mechanics, algebra, and the history of mathematics.
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34, No. 4, pp. 342–353, (1998). and also such special integrals as the integral of probability, the Fresnel integrals, the integral exponential function, the trigonometric integrals, and some other integralsE. A. Karatsuba, Fast computation of some special integrals of mathematical physics.
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Their measurements demonstrated an oblate Earth, with a flattening of 1:210. This approximation to the true shape of the Earth became the new reference ellipsoid. In 1787 the first precise trigonometric survey to be undertaken within Britain was the Anglo-French Survey.
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The Gauss–Legendre algorithm or Salamin–Brent algorithm was discovered independently by Richard Brent and Eugene Salamin in 1975. This can compute \pi to N digits in time proportional to N\,\log(N)\,\log(\log(N)), much faster than the trigonometric formulae.
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The Ailles rectangle The Ailles rectangle is a rectangle constructed from four right-angled triangles which is commonly used in geometry classes to find the values of trigonometric function of 15° and 75°. It is named after high school teacher Douglas S. Ailles.
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The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature.
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The geodetic infrastructure in New South Wales incorporates approximately 6,000 traditional trigonometric stations that formed the backbone of the survey control network before the introduction of more than 160 CORSnet-NSW stations. TP5865 - Collaroy Trig station is located on top of the heritage Collaroy Reservoir.
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Hobby Engineering: Aluminum Girder 210 mm from fischertechnik Other companies make Fischertechnik-compatible aluminum bars of any desired length.Staudinger: Fischertechnik-compatible aluminum profiles To teach the physics of such models, some sets included measuring devices, so that trigonometric vectors could be calculated and tested.
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The above integral may be expressed as an infinite truncated series by expanding the integrand in a Taylor series, performing the resulting integrals term by term, and expressing the result as a trigonometric series. In 1755, Euler derived an expansion in the third eccentricity squared.
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This family of maps {PN} is equicontinuous and tends to the identity on the dense subset consisting of trigonometric polynomials. It follows that PN f tends to f in Lp-norm for every . In other words, {xn} is a Schauder basis of Lp([0, 2π]).
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Stellar parallax motion from annual parallax. Half the apex angle is the parallax angle. The most important fundamental distance measurements come from trigonometric parallax. As the Earth orbits the Sun, the position of nearby stars will appear to shift slightly against the more distant background.
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Werner Crater on the moon The crater Werner on the Moon is named after him. Some of the trigonometric identities used in prosthaphaeresis, an early method for rapid computation of products, were named Werner formulas in honor of Werner's role in development of the algorithm.
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Since this definition works for any complex-valued z, this definition allows for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.
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Al-Khwārizmī's Zīj al-Sindhind also contained tables for the trigonometric functions of sines and cosine. A related treatise on spherical trigonometry is also attributed to him. Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.Jacques Sesiano, "Islamic mathematics", p.
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Exact trigonometric table for multiples of 3 degrees. Values outside the [0°, 45°] angle range are trivially derived from these values, using circle axis reflection symmetry. (See List of trigonometric identities.) In the entries below, when a certain number of degrees is related to a regular polygon, the relation is that the number of degrees in each angle of the polygon is (n – 2) times the indicated number of degrees (where n is the number of sides). This is because the sum of the angles of any n-gon is 180° × (n – 2) and so the measure of each angle of any regular n-gon is 180° × (n – 2) ÷ n.
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Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians.
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Algebraic number: Any number that is the root of a non- zero polynomial with rational coefficients. Transcendental number: Any real or complex number that is not algebraic. Examples include and . Trigonometric number: Any number that is the sine or cosine of a rational multiple of pi.
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Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions. Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics and communications, among other fields.
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She was born Cäcilia Wendt on May 4, 1875 in Troppau, Silesia. She studied at the University of Vienna from 1896 to 1900, where she published work on rational values of trigonometric functions, receiving a doctoral degree for research on special functions of importance in mathematical physics.
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In mathematics, Bhaskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I (c. 600 – c. 680), a seventh- century Indian mathematician. This formula is given in his treatise titled Mahabhaskariya.
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DENIS 0255−4700 was identified for the first time as a probable nearby object in 1999. Its proximity to the Solar System was established by the RECONS group in 2006 when its trigonometric parallax was measured. DENIS 0255-4700 has a relatively small tangential velocity of .
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Because of the Pythagorean trigonometric identity, the absolute value of \cos y + i \sin y is . Therefore, the real factor e^x is the absolute value of e^z and the imaginary part y of the exponent identifies the argument (angle) of the complex number e^z.
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Johnson was born in India to an Ordinance officer of the East India Company, who lived in "Deyrah". He was educated at Mussorie and joined the Civil Branch of the Great Trigonometric Survey (the precursor of the Survey of India), where he was trained by Andrew Scott Waugh.
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It has an instruction page, which explains the use of the plotter and the function syntax. About 180 functions are predefined. These belong to the categories basic functions, trigonometric and hyperbolic functions, non-differentiable functions, probability functions, special functions, programmable functions, iterations and fractals, differential and integral equations.
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Other fields that use trigonometry or trigonometric functions include music theory, geodesy, audio synthesis, architecture, electronics, biology, medical imaging (CT scans and ultrasound), chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, image compression, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
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Elementary Functions- a study of the elementary functions (power functions, polynomials, rational, exponential, logarithmic and trigonometric) with an emphasis on their behavior and applications. Some analytic geometry and elements of the calculus as well as the application of matrices to the solution of linear systems is also included.
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His lasting achievement is found in his book Geometria rotundi (1583), in which he introduced the modern names of the trigonometric functions tangent and secant. Geometriae rotundi libri XIIII, 1583 His son in law was the Danish physician and natural historian, Ole Worm, who married Fincke's daughter Dorothea.
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The Stoner–Wohlfarth model was developed by Edmund Clifton Stoner and Erich Peter Wohlfarth and published in 1948. It included a numerical calculation of the integrated response of randomly oriented magnets. Since this was done before computers were widely available, they resorted to trigonometric tables and hand calculations.
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In trigonometry, the trigonometric functions, such as \sin, \cos, and \tan, are unary operations. This is because it is possible to provide only one term as input for these functions and retrieve a result. By contrast, binary operations, such as addition, require two different terms to compute a result.
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In mathematics, a closed-form expression is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration.
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Solution of triangles () is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
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In the classical central-force problem of classical mechanics, some potential energy functions V(r) produce motions or orbits that can be expressed in terms of well-known functions, such as the trigonometric functions and elliptic functions. This article describes these functions and the corresponding solutions for the orbits.
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In addition to the law of sines, the law of cosines, the law of tangents, and the trigonometric existence conditions given earlier, for any triangle :a=b\cos C+c\cos B, \quad b=c\cos A+a\cos C, \quad c=a\cos B+b\cos A.
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Series are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function); etc.
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The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles. Amongst the lay public of non- mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics.
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Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis. When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
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The people of this civilization made bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure. They also tried to standardize measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. Indian astronomer and mathematician Aryabhata (476–550), in his Aryabhatiya (499) introduced a number of trigonometric functions (including sine, versine, cosine and inverse sine), trigonometric tables, and techniques and algorithms of algebra.
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In turn, Craig, who had studied with Wittich, accused Tycho of minimizing Wittich's role in developing some of the trigonometric methods used by Tycho. In his dealings with these disputes, Tycho made sure to leverage his support in the scientific community, by publishing and disseminating his own answers and arguments.
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Swingfield church was part of a chain of measuring points for the trigonometric survey linking the Royal Greenwich Observatory and the Paris Observatory in the late eighteenth century. This Anglo-French Survey was led by General William Roy, and used cross-channel sightings from nearby Dover Castle and Fairlight on the South Downs.
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In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of its derivatives. Informally speaking, the smoother the function is, the better it can be approximated by polynomials.
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In addition to general purpose calculators, there are those designed for specific markets. For example, there are scientific calculators which include trigonometric and statistical calculations. Some calculators even have the ability to do computer algebra. Graphing calculators can be used to graph functions defined on the real line, or higher-dimensional Euclidean space.
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Postnikov On character sums modulo a prime power, Izvestia Akad. Nauka SSSR, Ser. Math. 19, 1955, 11–16 This was also the subject of his Russian doctorate (higher doctoral recognition) in 1956 (Investigation of the method of Vinogradov for trigonometric sums (in Russian)). He was later a senior scientist at the Steklov Institute.
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The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,Katz, V. J. 1995. "Ideas of Calculus in Islam and India." Mathematics Magazine (Mathematical Association of America), 68(3):163-174. though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
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A Riemann surface for the argument of the relation . The orange sheet in the middle is the principal sheet representing . The blue sheet above and green sheet below are displaced by and respectively. Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complex plane.
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This clause is new; it recommends fifty operations, including log, power, and trigonometric functions, that language standards should define. These are all optional (none are required in order to conform to the standard). The operations include some on dynamic modes for attributes, and also a set of reduction operations (sum, scaled product, etc.).
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The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (continuous) function can be represented by a trigonometric series.
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A binary operation takes two arguments x and y, and returns the result x\circ y. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation.
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Graphical illustration of an elliptic function where its values are indicated by colours. These are periodically repeated in the two directions of the complex plane. Abel elliptic functions are holomorphic functions of one complex variable and with two periods. They were first established by Niels Henrik Abel and are a generalization of trigonometric functions.
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In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series.
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The whole structure is a fine example of riveted steel design and technology, surviving intact for more than 85 years. It is a prominent local landmark. Standard features include: davit, trigonometric station, access stairway, handrails and inlet and outlet valve chambers. Adjacent to the reservoir is a storage tank constructed with riveted steel plates.
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Krieger earned a B.A in 1924 and a M.A in 1925 from the University of Toronto. She obtained her Ph.D from the same university in 1930. Her thesis, under the supervision of W.J. Webber, was entitled "On the summability of trigonometric series with localized parameters—on Fourier constants and convergence factors of double Fourier series".
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Still, combined with trigonometric evidence of the form used by Eratosthenes 1,700 years prior, the Magellan expedition removed any reasonable doubt in educated circles in Europe. The Transglobe Expedition (1979–1982) was the first expedition to make a circumpolar circumnavigation, traveling the world "vertically" traversing both of the poles of rotation using only surface transport.
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In integral calculus, the Weierstrass substitution or tangent half-angle substitution is a method for evaluating integrals, which converts a rational function of trigonometric functions of x into an ordinary rational function of t by setting t = \tan (x /2).Weisstein, Eric W. "Weierstrass Substitution." From MathWorld--A Wolfram Web Resource. Accessed April 1, 2020.
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An unsealed road goes from Bells Line of Road to a picnic area about from the junction. At this point, vehicular access comes to an end. From the picnic area, a walking track makes its way approximately to the top of Mount Banks, where there is a trigonometric station. This is above sea level.
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In all the formulas stated below the sides , , and must be measured in absolute length, a unit so that the Gaussian curvature of the plane is −1. In other words, the quantity in the paragraph above is supposed to be equal to 1. Trigonometric formulas for hyperbolic triangles depend on the hyperbolic functions sinh, cosh, and tanh.
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Bass guitar time signal of open string A note (55 Hz). Fourier transform of bass guitar time signal of open string A note (55 Hz). Fourier analysis reveals the oscillatory components of signals and functions. In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
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Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented by means of complex contour integrals of products and quotients of the gamma function, called Mellin–Barnes integrals.
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Plumb bob lines descended from the scales above and intersected with linear scales marked on the horizontal scales below. These allowed measures to be read, not as angles, but as trigonometric ratios. To celebrate the 600th anniversary of the Rectangulus in 1926 a replica was constructed. This is now in the Museum of the History of Science, Oxford.
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Triangle with sides a,b,c and respectively opposite angles A,B,C Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities, relate both the sides and angles of a given triangle.
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There are several peaks on this mountain range. The tallest peak on the Kai Kung Leng mountain range is called Lo Tin Teng and is above sea level. Nearby, a peak simply called Kai Kung Leng, with the summit-signalling trigonometric post, stands at . Slightly further away to the west, a subpeak called Kai Kung Shan is tall.
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He may have used trigonometric methods that were available in his time, as he was a contemporary of Hipparchus. None of his original writings or Greek translations have survived, though a fragment of his work has survived only in Arabic translation, which was later referred to by the Persian philosopher Muhammad ibn Zakariya al-Razi (865-925).
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HU Delphini, also known as Gliese 791.2, is a star system in the constellation of Delphinus. Its apparent magnitude is 13.07. With a trigonometric parallax of 113.4 ± 0.2 mas, it is about 28.76 light-years (8.82 parsecs) away from the Solar System. HU Delphini is a binary star with a well-defined period of 538.6 days.
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A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic. An illustration of a periodic function with period P.
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In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. Divided differences is a recursive division process. The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.
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It is not known how Bhaskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhaskara might have used to arrive at his formula. The formula is elegant, simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever.
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Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions.
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Currently the most precise distance estimate of 2MASS 0937+2931 is trigonometric parallax, published in 2009 by Schilbach et al.: 163.39 ± 1.76 mas, corresponding to a distance 6.12 ± 0.07 pc, or 19.96 ± 0.22 ly. A less precise parallax of this object, measured under U.S. Naval Observatory Infrared Astrometry Program, was published in 2004 by Vrba et al.
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This function property leads to exponential growth or exponential decay. The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.
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In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typical exponential sum may take the form :\sum_n e(x_n), summed over a finite sequence of real numbers xn.
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Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular trigonometric functions by regarding a hyperbolic angle as defining a hyperbolic triangle. The parameter thus becomes one of the most useful in the calculus of real variables.
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Takebe played critical role in the development of the Enri (, "circle principle") - a crude analogon to the western calculus. He also created charts for trigonometric functions. Mathematical Society of Japan, Takebe Prize He obtained power series expansion of (\arcsin(x))^2 in 1722, 15 years earlier than Euler. This was the first power series expansion obtained in Wasan.
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VSIPL is an application programming interface (API). VSIPL and VSIPL++ contain functions used for common signal processing kernel and other computations. These functions include basic arithmetic, trigonometric, transcendental, signal processing, linear algebra, and image processing. The VSIPL family of libraries has been implemented by multiple vendors for a range of processor architectures, including x86, PowerPC, Cell, and NVIDIA GPUs.
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The same law was also investigated by Ptolemy and in the Middle Ages by Witelo, but due to lack of adequate mathematical instruments (i.e. trigonometric functions) their results were saved as tables, not functions. The lunar crater Snellius is named after Willebrord Snellius. The Royal Netherlands Navy has named three survey ships after Snellius, including a currently-serving vessel.
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The Chebyshev method is a recursive algorithm for finding the th multiple angle formula knowing the th and th values. can be computed from , , and with :. This can be proved by adding together the formulae : :. It follows by induction that is a polynomial of , the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.
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These user-customizable software calculators can also be used in conjunction with formula or equation creation capabilities so that the software calculator can now be created to perform all possible mathematical functions. No longer limited to a set of trigonometric and simple algebraic expressions, versions of the software calculator are now tailored to any and all topical applications.
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Performance of trigonometric functions is bad compared to C, because Java has strict specifications for the results of mathematical operations, which may not correspond to the underlying hardware implementation. On the x87 floating point subset, Java since 1.4 does argument reduction for sin and cos in software, causing a big performance hit for values outside the range.
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Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers. Hipparchus of Nicaea (c. 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.
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Those that can be expressed in terms of square roots are well characterized (see Trigonometric constants expressed in real radicals). To express the other ones in terms of radicals, one requires th roots of non-real complex numbers, with . An elementary proof that every trigonometric number is an algebraic number is as follows.. One starts with the statement of de Moivre's formula for the case of \theta = 2\pi k/n for coprime k and n: :(\cos \theta + i \sin \theta )^n =1. Expanding the left side and equating real parts gives an equation in \cos \theta and \sin^2 \theta; substituting \sin^2 \theta =1-\cos^2 \theta gives a polynomial equation having \cos \theta as a solution, so by definition the latter is an algebraic number.
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'''''''''' The astronomical treatise Āryabhaṭīya was composed during the fifth century by the Indian mathematician and astronomer Āryabhaṭa (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns. Arc and chord of a circle Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as Āryabhaṭa's table of sine-differences. Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.
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Winckel, Fritz (1967). Music, Sound and Sensation: A Modern Exposition, p.134. Courier. . If a graph is drawn to show the function corresponding to the total sound of two strings, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of one wave cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume. It can be proven with the help of a sum-to-product trigonometric identity (see List of trigonometric identities) that the envelope of the maxima and minima form a wave whose frequency is half the difference between the frequencies of the two original waves.
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In 1592 Magini published Tabula tetragonica, and in 1606 devised extremely accurate trigonometric tables. He also worked on the geometry of the sphere and applications of trigonometry, for which he invented calculating devices. He also worked on the problem of mirrors and published on the theory of concave spherical mirrors. He also published a commentary on Ptolemy’s Geographia (Cologne, 1596).
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His starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. The year after Abel could report that his new functions had two periods.O. Ore, Niels Henrik Abel – Mathematician Extraordinary, AMS Chelsea Publishing, Providence, RI (2008). . Especially this property made them more interesting than the normal trigonometric functions which have only one period.
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The interpreter supported a full set of scientific functions (trigonometric functions, logarithm etc.) at this accuracy. The language supported two-dimensional arrays, and a ROM extension made high-level functions such as matrix multiplication and inversion available. For the larger HP-86 and HP-87 series, HP also offered a plug-in CP/M processor card with a separate Zilog Z-80 processor.
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By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well. For the history of such tables, see log table.
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Later distance estimates of the star, besides trigonometric parallax with high uncertainty from the star's discovery paper,() include a parallax of "~128 mas" without specific error range from Burgasser et al. (2015). Its cross- references, including for parallax, were the 2005 discovery paper and T. Henry, priv. comm. In independent agreement with the latter, Gaia's Data Release 2 gives a parallax of .
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Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ2. The Haar system is an example of a basis for Lp([0, 1]), when 1 ≤ p < ∞. When , another example is the trigonometric system defined below. The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the supremum norm, admits a Schauder basis.
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Like all deterministic global optimization software, ANTIGONE requires the user to provide the explicit mathematical expressions for all the functions used in the problem, as well as initial bounds for all variables. If initial bounds are not supplied, ANTIGONE will attempt to infer bounds, but global optimality is not guaranteed. ANTIGONE can only solve differentiable functions, and can not solve trigonometric problems.
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Accessed on line July 10, 2010. An initial photometric estimate for the distance to 2M1207b was 70 parsecs. In December 2005, American astronomer Eric Mamajek reported a more accurate distance (53 ± 6 parsecs) to 2M1207b using the moving-cluster method. Recent trigonometric parallax results have confirmed this moving-cluster distance, leading to a distance estimate of 52.75 parsecs or 172 ± 3 light years.
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Armadillo is a linear algebra software library for the C++ programming language. It aims to provide efficient and streamlined base calculations, while at the same time having a straightforward and easy-to-use interface. Its intended target users are scientists and engineers. It supports integer, floating point (single and double precision), complex numbers, and a subset of trigonometric and statistics functions.
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Georg Joachim de Porris, also known as Rheticus (/ˈrɛtɪkəs/; 16 February 1514 – 4 December 1574), was a mathematician, astronomer, cartographer, navigational-instrument maker, medical practitioner, and teacher. He is perhaps best known for his trigonometric tables and as Nicolaus Copernicus's sole pupil.Danielson, p. 3. He facilitated the publication of his master's De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres).
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The Nalaknad (also known as Nalnad - meaning 4 villages) palace at the foothills is an important historical landmark. This was one of the landmarks mapped during the Great Trigonometric Survey. It is a place of interest for trekkers and naturalists. The climb to the top and back can be completed as a day hike; camping is banned since December 2016.
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However, for trigonometric and hyperbolic functions, this notation conventionally means exponentiation of the result of function application. The expression a/2b can be interpreted as meaning (a/2)b, in particular if one thinks that the common acronym PEMDAS for the order of operations implies that M(ultiplication) takes precedence over D(ivision); however, it is more commonly understood to mean a/(2b).
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Applying his p-adic form of the Hardy- Littlewood-Ramanujan-Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Karatsuba obtained a new estimate of the well known Hardy function G(n) in the Waring's problem (for n \ge 400): : \\! G(n) < 2 n\log n + 2 n\log\log n + 12 n.
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The 2nd edition of Zygmund's Trigonometric Series (Cambridge University Press, 1959) consists of 2 separate volumes. The 3rd edition (Cambridge University Press, 2002, ) consists of the two volumes combined with a foreword by Robert A. Fefferman. The nine pages in Fefferman's foreword (biographic and other information concerning Zygmund) are not numbered. Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst.
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In 1689, French soldiers finally destroyed the place. The surviving keep, or bergfried, was initially used as a prison, but was no longer fit for that purpose after 1752. The castle was abandoned and used as a stone quarry. In 1818 Prussia had the bergfried restored because, with its height of , it would be able to act as a trigonometric point.
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Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.
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In Kresa's era the Trigonometric functions were derived using geometry. Kresa was the first to introduce algebraic number to trigonometry. Kresa's death was followed by a decline in mathematics and science in the Czech Crown lands due to the dogmatic application of Catholic Church doctrines. With Slavíček having gone to China, scientific work largely disappeared from the Czech lands for two decades.
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In 1982 Wilhelm Gliese published photometric distance of LHS 1723 (161 mas), and in 1991 it was included to the 3rd preliminary version of catalogue of nearby stars by Gliese and Jahreiss as NN 3323 (also designated as GJ 3323) with photometric parallax . Its trigonometric parallax remained unknown until 2006, when it was published by RECONS team. The parallax was .
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But, as advanced as they were, they attributed no refraction whatever above 45° altitude for solar refraction, and none for starlight above 20° altitude. To perform the huge number of multiplications needed to produce much of his astronomical data, Tycho relied heavily on the then-new technique of prosthaphaeresis, an algorithm for approximating products based on trigonometric identities that predated logarithms.
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Electronic calculators began to be owned at school from the early 1980s, becoming widespread from the mid-1980s. Parents and teachers believed that calculators would diminish abilities of mental arithmetic. Scientific calculators came to the aid for those working out logarithms and trigonometric functions. Since 1988, exams in Mathematics at age sixteen, except Scotland, have been provided by the GCSE.
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A primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions). The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series (see ). In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit, which has been described as the first formula for the DFT, and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string. Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.
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For example, the graph of y = x − 4x + 7 can be obtained from the graph of y = x by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2). For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = ), then stretching it parallel to the Y axis using a stretch factor R, where R = A + B. This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities).
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The other standard method of levelling in construction and surveying is called "trigonometric levelling", which is preferred when levelling "out" to a number of points from one stationary point. This is done by using a total station, or any other instrument to read the vertical, or zenith angle to the rod, and the change in elevation is calculated using the sine trigonometric function (see example below). At greater distances (typically 1,000 feet and greater), the curvature of the Earth, and the refraction of the instrument wave through the air must be taken into account in the measurements as well (see section below). Ex: an instrument reading to a rod a zenith angle of < 48°15'22" (degrees, minutes, seconds of arc) and a slope distance of 305.50 feet would be calculated thus: sin(48°15'22")(305.5)= 227.95 ft.
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A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions and cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric functions and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions, which are the functions which may be built from rational functions, roots of a polynomial, logarithm, and exponential functions. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule.
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Omicron Scorpii was occasionally mentioned as a possible member of the Upper Scorpius sub-group in the Scorpius-Centaurus OB association during the 20th century. However, it does not appear in more recent membership lists for this group due to its small proper motion and small trigonometric parallax as measured by Hipparcos. This suggests that it is a background star unrelated to Scorpius- Centaurus.
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Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this problem in 1869.
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Some of these methods used tables derived from trigonometric identities.Enrique Gonzales-Velasco (2011) Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, Springer Such methods are called prosthaphaeresis. Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague.
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Bianchini was the first mathematician in Europe to use decimal positional fractions for his trigonometric tables, at the same time as Al-Kashi in Samarkand. In De arithmetica, part of the Flores almagesti, he uses operations with negative numbers and expresses the Law of Signs. He was probably the father of the instrument maker Antonio Bianchino. The crater Blanchinus on the Moon is named after him.
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In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity :\cos\alpha\cos\beta = \frac12[\cos(\alpha+\beta) + \cos(\alpha-\beta)] or similar to convert the multiplications to additions and table lookups. However, logarithms are more straightforward and require less work. It can be shown using Euler's formula that the two techniques are related.
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A page from Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.
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Since the common logarithm of a power of is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For numbers less than the characteristic makes the resulting logarithm negative, as required.E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). See common logarithm for details on the use of characteristics and mantissas.
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He served as chairman of the astronomy department at Wesleyan, then in 1960 he was named as emeritus Fisk professor of astronomy. From 1960–71, after serving as an assistant, he became director of the Van Vleck Observatory; the second to hold that position. During his career he computed more than 200 stellar trigonometric parallaxes. In 1927 he discovered the comet 1927 IV (comet Stearns, 1927d).
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Light, who was slowly succumbing to tuberculosis, managed to complete by December 1837, by which time the population had increased to around 2,500. When Kingston returned in June 1838, had been completed. Light's requests were denied; instead he could change from the trigonometric surveys to a faster (but inferior) running survey, or hand control over to Kingston and confine himself to coastal surveys. Light resigned in protest.
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Trigonometric parallax of 2MASS 1114−2618, measured in 2012 by Dupuy & Liu under The Hawaii Infrared Parallax Program, is 0.1792 ± 0.0014 arcsec, corresponding to a distance 5.58 ± 0.04 pc, or 18.20 ± 0.14 ly. Photometric distance estimate of 2MASS 1114−2618, published in its discovery paper in 2005, is 7 pc (22.8 ly). Spectrophotometric distance estimate by Kirkpatrick et al. (2012), is 6.6 pc (21.5 ly).
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Mathematician and cryptographer Neal Koblitz was one of Katz's students. Katz studied, with Sarnak among others, the connection of the eigenvalue distribution of large random matrices of classical groups to the distribution of the distances of the zeros of various L and zeta functions in algebraic geometry. He also studied trigonometric sums (Gauss sums) with algebro-geometric methods. He introduced the Katz–Lang finiteness theorem.
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During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions. Jyeṣṭhadeva provided proofs for these infinite series in the Yuktibhāṣā.Katz, V. J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Mathematical Association of America) 68 (3): 163-74.
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See also for various other examples in degree 5. Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals. See Radical extension for the precise formulation of his result. Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.
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His mathematical works were in the areas of spherical trigonometry, as well as conic sections. He published an original work on conic sections in 1522 and is one of several mathematicians sometimes credited with the invention of prosthaphaeresis, which simplifies tedious computations by the use of trigonometric formulas, sometimes called Werner's formulas.Howard Eves, An Introduction to the History of Mathematics, Sixth Edition, p. 309, Thompson, 1990, .
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Kharkiv Observatory in 1914. A few years after retirement of his father, in 1894, Ludwig moved to the University of Kharkov. There, in 1897 he became professor in astronomy and geodesy and director of the observatory. Prior to Struve, the Kharkov Observatory was not registered within the Russian leveling network and the altitude of Kharkov was based on rather inaccurate trigonometric leveling conducted by local triangulation.
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The conclusion is that, since is the same as and is the same as , it is true that and . It may be inferred in a similar manner that , since and . A simple demonstration of the above can be seen in the equality . When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than .
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Interest in the properties of optics, and light, date back to almost 2000 years to Ptolemy (AD 85 – 165). In his work entitled Optics, he writes about the properties of light, including reflection, refraction, and color. He developed a simplified equation for refraction without trigonometric functions. About 800 years later, in AD 984, Ibn Sahl discovered a law of refraction mathematically equivalent to Snell's law.
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Windows Calculator is a software calculator developed by Microsoft and included in Windows. It has four modes: standard, scientific, programmer, and a graphing mode. The standard mode includes a number pad and buttons for performing arithmetic operations. The scientific mode takes this a step further and adds exponents and trigonometric function, and programmer mode allows the user to perform operations related to computer programming.
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Robert of Chester's translations into Latin included al-Khwarizmi's Algebra and astronomical tables (also containing trigonometric tables). Abraham of Tortosa's translations include Ibn Sarabi's (Serapion Junior) De Simplicibus and Abulcasis' Al-Tasrif as Liber Servitoris. In 1126, Muhammad al-Fazari's Great Sindhind (based on the Sanskrit works of Surya Siddhanta and Brahmagupta's Brahmasphutasiddhanta) was translated into Latin.G. G. Joseph, The Crest of the Peacock, p. 306.
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The French artillery has used the grad for decades. Today, the degree, of a turn, or the mathematically more convenient radian, of a turn (used in the SI system of units) are generally used instead. In the 1970s - 1990s, most scientific calculators offered the grad, as well as radians and degrees, for their trigonometric functions. In the 2010s, some scientific calculators lack support for gradians.
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His mathematical work included works on the determinant, hyperbolic functions, and parabolic logarithms and trigonometry.This is about connecting the rectified length of line segments along a parabola, giving logarithms for appropriate coordinates, and trigonometric values for suitable angles, in a similar way as the area under a hyperbola defines the natural logarithm, and a hyperbolic angle is defined via the area of a hyperbolically truncated triangle.
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For these applications, some small processors feature BCD arithmetic modes, which assist when writing routines that manipulate BCD quantities. Where calculators have added functions (such as square root, or trigonometric functions), software algorithms are required to produce high precision results. Sometimes significant design effort is needed to fit all the desired functions in the limited memory space available in the calculator chip, with acceptable calculation time.
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Using the trigonometric terms for a right triangle, specifically opposite, adjacent, and hypotenuse, the adjacent side was fixed by construction. One variable changed the magnitude of the opposite side. In many cases, this variable changed sign; the hypotenuse could coincide with the adjacent side (a zero input), or move beyond the adjacent side, representing a sign change. Typically, a pinion- operated rack moving parallel to the (trig.
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The latter were translated by Gerard of Cremona. Nairizi used the so-called umbra (versa), the equivalent to the tangent, as a genuine trigonometric line (but he was anticipated in this by al-Marwazi). He wrote a treatise on the spherical astrolabe, which is very elaborate and seems to be the best Persian work on the subject. It is divided into four books: #Historical and critical introduction.
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The configurations correspondingly responsible for higher, i.e. excited, states are periodic instantons defined on a circle of Euclidean time which in explicit form are expressed in terms of Jacobian elliptic functions (the generalization of trigonometric functions). The evaluation of the path integral in these cases involves correspondingly elliptic integrals. The equation of small fluctuations about these periodic instantons is a Lamé equation whose solutions are Lamé functions.
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The first, in 1991, gave a parallax of , yielding a distance of roughly or . The second was the Hipparcos Input Catalogue (1993) with a trigonometric parallax of , a distance of or . Given this uncertainty, researchers were adopting a wide range of distance estimates, leading to significant variances in the calculation of the star's attributes. The results from the Hipparcos mission were released in 1997.
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In Prolog for example, the infix operator is non-associative, so constructs such as are syntax errors. Unary prefix operators such as − (negation) or sin (trigonometric function) are typically associative prefix operators. When more than one associative prefix or postfix operator of equal precedence precedes or succeeds an operand, the operators closest to the operand goes first. So −sin x = −(sin x), and sin -x = sin(-x).
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There he calculated extensive astronomical tables and built astronomical instruments. Next he went to Buda, and the court of Matthias Corvinus of Hungary, for whom he built an astrolabe, and where he collated Greek manuscripts for a handsome salary. The trigonometric tables that he created while living in Hungary, his Tabulae directionum profectionumque (printed posthum., 1490), were designed for astrology, including finding astrological houses.
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The functionality of calculator ICs increased at a rapid pace and Roberts was designing and producing new models. The MITS 7400 scientific and engineering calculator was introduced in December 1972. It featured trigonometric functions, polar to rectangular conversion, two memories, and up to a seven-level stack. A kit with a three- level stack was $299.95 and an assembled unit with a seven-level stack was $419.95.
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Currently the most precise distance estimate of 2MASS 0415−0935 is published in 2012 by Dupuy & Liu trigonometric parallax, measured under The Hawaii Infrared Parallax Program: 175.2 ± 1.7 milliseconds of arc, corresponding to a distance 5.71 ± 0.05 pc, or 18.62 ± 0.18 ly. A less precise parallax of this object, measured under U.S. Naval Observatory Infrared Astrometry Program, was published in 2004 by Vrba et al.
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The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name—for example, = . Nevertheless, certain authors advise against using it for its ambiguity. Another convention used by a few authors is to use an uppercase first letter, along with a superscript: , , , etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by , , etc.
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Kiama Reservoir, located at the south end of the site, is a simple circular reinforced concrete reservoir with a thickened upper rim or rib. There is no visible concrete apron. Standard features include a handrail in tubular steel, davit, access ladder, inlet and outlet valve chambers and a trigonometric station. It has a full service level of 90 m and a capacity of 1.1ML.
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From the picnic area, a track goes to the mountain, which is just over a kilometre away. There are good views of the Grose Valley as the track approaches the mountain, but the views disappear as the track gets higher up. At the top, there is a trigonometric station, but there are no views because of the timber. The top of the mountain is above sea level.
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Illustration from Bellicorum instrumentorum liber, Venice c. 1420 - 1430 Fontana composed treatises on a diverse array of topics, including measurement of heights or depths by falling stones. We have early works of his on water- clocks (with wheels), sand-clocks and measurement. Fontana studied trigonometric measurements, mentioned in De trigono balistario, and through his own designed instrument, also explained in a larger treatise, however, apparently lost.
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Depending on the initial conditions, even with all roots real the iterates can experience a transitory tendency to go above and below the steady state value. But true cyclicity involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate characteristic roots. This can be seen in the trigonometric form of their contribution to the solution equation, involving and .
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The translations and reflections are ones that correspond to the symmetries and periodicities of the basic trigonometric functions. Bioche's rules state that: # If \omega(-t)=\omega(t), a good change of variables is u=\cos t. # If \omega(\pi-t)=\omega(t), a good change of variables is u=\sin t. # If \omega(\pi+t)=\omega(t), a good change of variables is u=\tan t.
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Mount Dorothy Reservoir is a cylindrical prestressed concrete reservoir, built using pre cast panels wrapped around with high tensile steel wire. It has a diameter of 39m and is 7.5m deep. The reservoir has a capacity of 9.3 ML. It is similar in construction to Cecil Park Reservoir (WS 165). Standard features include: concrete apron, davit, trigonometric station, access ladder, handrails and inlet and outlet valve chambers.
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This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest. Let \theta_3=90^\circ. Then \theta_1+(\theta_2+\theta_4)=90^\circ . Hence : \sin\theta_1\sin\theta_3+\sin\theta_2\sin\theta_4=\sin(\theta_3+\theta_2)\sin(\theta_3+\theta_4) : \cos(\theta_2+\theta_4)+\sin\theta_2\sin\theta_4=\cos\theta_2\cos\theta_4 : \cos(\theta_2+\theta_4)=\cos\theta_2\cos\theta_4-\sin\theta_2\sin\theta_4 Formula for compound angle cosine (+) Despite lacking the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem (or more simply the Second Theorem) the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cognoscenti of those times to draw up accurate tables of chords (corresponding to tables of sines) and to use these in their attempts to understand and map the cosmos as they saw it.
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For example, . The sequence of double factorials for starts as : 1, 3, 15, 105, 945, , ,... Double factorial notation may be used to simplify the expression of certain trigonometric integrals, to provide an expression for the values of the gamma function at half-integer arguments and the volume of hyperspheres,. and to solve many counting problems in combinatorics including counting binary trees with labeled leaves and perfect matchings in complete graphs..
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The new reduction of the Hipparcos data gave 0.12 ± 14.62 milliarcseconds, still unusable. The General Catalogue of Trigonometric Parallaxes, an older catalogue of ground-based parallaxes, lists the parallax as 20 ± 16 milliarcseconds, corresponding to about . Beta Phoenicis is a relatively wide visual binary consisting of two G-type giant stars, both with spectral types of G8III. The two orbit each other every 170.7 years and have a relatively eccentric orbit.
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In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see . For the case of roots of unity in rings of modular integers, see Root of unity modulo n.
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Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.
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Originally the most precise distance estimate of 2MASS 1503+2525 is a trigonometric parallax, published by Dupuy and Liu in 2012: 157.2 ± 2.2 mas, corresponding to a distance 6.36 ± 0.09 pc, or 20.7 ± 0.3 ly. The parallax was further refined by Gaia mission in 2018 to 154.9208mas. The brown dwarf 2MASS 1503+2525 lies in local void 6.5 parsecs across, where relatively few stars and brown dwarfs are located.
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Harmukh was first climbed by the Great Trigonometric Survey's Thomas Montgomerie in 1856 and made the first survey of the Karakoram some to the south, and sketched the two most prominent peaks, labelling them K1 and K2. Harmukh was later climbed by many other climbers. Therefore, Harmukh is the mountain from which the world's 2nd highest mountain peak K2 was discovered and the Serveyer's mark K2 continues to be the name.
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The tsunami was recorded by eleven of a series of continuous tide gauges around the Bay of Bengal that had recently been deployed by the Great Trigonometric Survey of India. The ten gauges on the Indian mainland were synchronised using a telegraph to Madras (Chennai) time, while that at Port Blair was set by a chronometer linked to local time. The maximum recorded wave height was at Nagapattinam.
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In analog computers, high precision potentiometers are used to scale intermediate results by desired constant factors, or to set initial conditions for a calculation. A motor-driven potentiometer may be used as a function generator, using a non-linear resistance card to supply approximations to trigonometric functions. For example, the shaft rotation might represent an angle, and the voltage division ratio can be made proportional to the cosine of the angle.
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Oblique vector rectifies to Slerp factor. More familiar than the general Slerp formula is the case when the end vectors are perpendicular, in which case the formula is . Letting , and applying the trigonometric identity , this becomes the Slerp formula. The factor of in the general formula is a normalization, since a vector p1 at an angle of Ω to p0 projects onto the perpendicular ⊥p0 with a length of only .
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He was able to discredit an opposing 'expert' because he'd misread some trigonometric tables. Cook's lawyers won, and gave him $500 for this day in court. He was asked to help in other cases, but Bill Mussen objected to this, so Cook withdrew. To maintain his 'second profession' of teaching, Cook taught math in night school at the San Antonio College for two years, including trig, analytical geometry and calculus.
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The Habsburgwarte that stands atop the Hermannskogel was established as the kilometre zero of cartographic measurements in Austria-Hungary at the start of the 19th century. In the 1920s however, Austria adopted the Gauss–Krüger coordinate system. Thereafter, the Hermannskogel had the same function as a trigonometric reference point as the Rauenberg point on the Marienhöhe in Berlin. The transition to the 1989 European Territorial Reference System will take place soon.
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The reference level was the height of Mean Water at Liverpool. 32 end points were connected to tidal stations and therefore the variation of mean water around the country was fixed. The benchmarking proceeded along the roads, but side lines were taken to, and over, many of the mountain top trig points, from which other trig point altitudes were measured by trigonometric levelling. The reports were published in 1861.
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Jacques Ozanam was born in Sainte- Olive, Ain, France. In 1670, he published trigonometric and logarithmic tables more accurate than the existing ones of Ulacq, Pitiscus, and Briggs. An act of kindness in lending money to two strangers brought him to the attention of M. d'Aguesseau, father of the chancellor, and he secured an invitation to settle in Paris. There he enjoyed prosperity and contentment for many years.
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This is done in the same manner as when one uses a transit using a set of trigonometric formulae based on height and angle. The Relascope is not commonly used for this because of its difficulty and the amount of time it takes to do this. Tree height is another use of this instrument. It does this by using several weighted wheels that spin based on the position of the instrument.
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Other types of curves, such as trigonometric functions (such as sine and cosine), may also be used, in certain cases. In spectroscopy, data may be fitted with Gaussian, Lorentzian, Voigt and related functions. In agriculture the inverted logistic sigmoid function (S-curve) is used to describe the relation between crop yield and growth factors. The blue figure was made by a sigmoid regression of data measured in farm lands.
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Great Trigonometric Survey ridge stone present in Bisle village, Karnataka Triangulation surveys were based on a few carefully measured baselines and a series of angles. The initial baseline was measured with great care since the accuracy of the subsequent survey was critically dependent upon it. Various corrections were applied, principally temperature. An especially accurate folding chain was used, laid on horizontal tables, all shaded from the sun and with constant tension.
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For some mathematical functions, a gold "f−1" prefix key would access the inverse of the gold-printed functions, e.g. "f−1" followed by "4" would calculate the inverse sine (sin^{-1}). Functions included square root, inverse, trigonometric (sine, cosine, tangent and their inverses), exponentiation, logarithms and factorial. The HP-65 was one of the first calculators to include a base conversion function, although it only supported octal (base 8) conversion.
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In 1856, Thomas George Montgomerie, a member of the British Royal Engineers and part of the Great Trigonometric Survey, sighted the mountain and named it "K4", meaning the fourth mountain of Karakoram. The name "Gasherbrum" comes from the Balti words rgasha ("beautiful") and brum ("mountain"); it does not, contrary to popular belief, mean "shining wall", how Sir William Martin Conway described nearby Gasherbrum IV on an 1892 exploration.
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The classic example is truncated power series representations of sin(x) and related trigonometric functions. For instance, taking only data from near the x = 0, we may estimate that the function behaves as sin(x) ~ x. In the neighborhood of x = 0, this is an excellent estimate. Away from x = 0 however, the extrapolation moves arbitrarily away from the x-axis while sin(x) remains in the interval [−1,1]. I.e.
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They named it WISEA J154045.67-510139.3 and assigned it spectral type M6. Pérez Garrido and colleagues were looking for high proper motion sources in the 2MASS–WISE cross-match. They named it 2MASS J154043.42−510135.7 (2M1540) and classified it as an M7.0±0.5 dwarf. Since the trigonometric distance of 2M1540 agreed with its spectrophotometric distances, computed for a single object, it was concluded that it is not an equal-mass binary.
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Without the advancements made in nautical sciences, particularly by Iberian scientists and explorers, trans-oceanic navigation would have not been possible. The earliest periods of navigation within Portugal and Spain employed the use of crude, antiquated, and unreliable instruments. The kamal and cross-staff, while both useful, lacked the ability to be consistently practically applied on board a ship. They were replaced by the trigonometric quadrant and its semi-circle construction.
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In 1861, Carter served in the Sikkim Expedition. In 1863, he commanded the engineer forces in the Umbeyla Campaign. On 21 April 1864, Carter was appointed a 3rd Grade surveyor in Great Trigonometric Survey of India. Later that year, on 15 September, he was married to Emily Georgina Campbell of Possil, the daughter of General George Campbell of Inverniell and he adopted the new name of Carter-Campbell of Possil.
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The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations. His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among the most significant were the results obtained with Calderón on singular integral operators. George G. Lorentz called it Zygmund's crowning achievement, one that "stands somewhat apart from the rest of Zygmund's work".
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Engraving, c. 1815 The tower was used by General William Roy in his trigonometric survey linking the nearby Royal Greenwich Observatory with the Paris Observatory; a 36 inch (0.91 m) theodolite (now in London's Science Museum) was temporarily installed on its roof. This Anglo-French Survey (1784–1790) led to the formation of the Ordnance Survey. In 1848, the Royal Engineers used the castle for their survey of London.
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This table was further extended by Jurij Vega in 1794, and by Alexander John Thompson in 1952. A shorter trigonometric table called Canon Sinuum was included in later works of Vlacq. In 1632, he settled in London but ten years later with the onset of the English Civil War, he moved to Paris and later moved to The Hague. He died at The Hague on 8 April 1667.
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In December 2005, American astronomer reported a more accurate distance (53 ± 6 parsecs) to 2M1207 using the moving cluster method. The new distance gives a fainter luminosity for 2M1207. Recent trigonometric parallax results have confirmed this moving cluster distance, leading to a distance estimate of 53 ± 1 parsec or 172 ± 3 light years."The Distance to the 2M1207 System" , Eric Mamajek, November 8, 2007. Accessed on line June 15, 2008.
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This is the special advantage of the form of table first introduced by Professor Inman, of the Portsmouth Royal Navy College, nearly a century ago.W. W. Sheppard and C. C. Soule, Practical navigation (World Technical Institute: Jersey City, 1922).E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). These days, the haversine form is also convenient in that it has no coefficient in front of the function.
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She was a close collaborator with Dmitrii Menshov on a number of research projects. She and Menshov took charge of function theory work at Moscow State during the 1940s. In 1952, she published an important piece on primitive functions, and trigonometric series and their almost everywhere convergence. Bari also posted works at the 1956 Third All- Union Congress in Moscow and the 1958 International Congress of Mathematicians in Edinburgh.
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The sine and cosine functions are one-dimensional projections of uniform circular motion. Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves. Under rather general conditions, a periodic function can be expressed as a sum of sine waves or cosine waves in a Fourier series.
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In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top. This yielded more accurate measurements of the Earth's circumference and made it possible for a single person to measure it from a single location.Lenn Evan Goodman (1992), Avicenna, p. 31, Routledge, .
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Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al- Battānī () (Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was an Arab astronomer, and mathematician. He introduced a number of trigonometric relations, and his Kitāb az-Zīj was frequently quoted by many medieval astronomers, including Copernicus. Often called the "Ptolemy of the Arabs",Barlow, Peter; Kater, Henry; Herschel, Sir John Frederick William (1856).
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Dmitrii Menshov studied languages as a schoolboy, but from the age of 13 he began to show great interest in mathematics and physics. In 1911, he completed high school with a gold medal. After a semester at the Moscow Engineering School, he enrolled at Moscow State University in 1912 and became a student of Nikolai Luzin. In 1916, Menshov completed his dissertation on the topic of trigonometric series.
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If needed, the interior angles of triangle C1-C2-P can be found using the trigonometric law of cosines. Also, if needed, the coordinates of P can be expressed in a second, better-known coordinate system—e.g., the Universal Transverse Mercator (UTM) system—provided the coordinates of C1 and C2 are known in that second system. Both are often done in surveying when the trilateration method is employed.
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Based on engraved plans of Meroitic King Amanikhabali's pyramids, Nubians had a sophisticated understanding of mathematics as they appreciated the harmonic ratio. The engraved plans is indicative of much to be revealed about Nubian mathematics. The ancient Nubians also established a system of geometry which they used in creating early versions of sun clocks. During the Meroitic period in Nubian history, the Nubians used a trigonometric methodology similar to the Egyptians.
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During the 18th century, great Trigonometric Survey is organised by Sir George Everest to measure the height of Mount Everest in which Sironj is one of the three observatories in India. This survey was later completed by Radhanath Sikdar using a theodolite. The ruins of these observatories still exist near Sironj, in a village called BHOORI TORI on Guna road. One can find there a good English architecture.
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The Cx4 coprocessor chip in Mega Man X2. The Cx4 chip is a math coprocessor that was used by Capcom and produced by Hitachi (now Renesas) to perform general trigonometric calculations for wireframe effects, sprite positioning and rotation. It is known for its role in mapping and transforming wireframes in Capcom's second and third Mega Man X series games. It is based on the Hitachi HG51B169 DSP and clocked at 20Mhz.
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In 1861, Carter served in the Sikkim Expedition. In 1863, he commanded the engineer forces in the Umbeyla Campaign. On 21 April 1864, Carter was appointed a 3rd Grade surveyor in Great Trigonometric Survey of India. Later that year, on 15 September, he was married to Emily Georgina Campbell of Possil, the daughter of General George Campbell of Inverniell and he adopted the new name of Carter-Campbell of Possil.
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The diagonal of the unit cube is . This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio . The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1. If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length and . From this the trigonometric function tangent of 60° equals , and the sine of 60° and the cosine of 30° both equal . The square root of 3 also appears in algebraic expressions for various other trigonometric constants, includingJulian D. A. Wiseman Sin and Cos in Surds the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.
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According to Henk Bos, :The Introduction is meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of the differential and integral calculus. [Euler] made of this survey a masterly exercise in introducing as much as possible of analysis without using differentiation or integration. In particular, he introduced the elementary transcendental functions, the logarithm, the exponential function, the trigonometric functions and their inverses without recourse to integral calculus — which was no mean feat, as the logarithm was traditionally linked to the quadrature of the hyperbola and the trigonometric functions to the arc-length of the circle.H. J. M. Bos (1980) "Newton, Leibnitz and the Leibnizian tradition", chapter 2, pages 49–93, quote page 76, in From the Calculus to Set Theory, 1630 – 1910: An Introductory History, edited by Ivor Grattan-Guinness, Duckworth Euler accomplished this feat by introducing exponentiation ax for arbitrary constant a in the positive real numbers.
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In 1868 Beltrami published two memoirs (written in Italian; French translations by J. Hoüel appeared in 1869) dealing with consistency and interpretations of non-Euclidean geometry of János Bolyai and Nikolai Lobachevsky. In his "Essay on an interpretation of non-Euclidean geometry", Beltrami proposed that this geometry could be realized on a surface of constant negative curvature, a pseudosphere. For Beltrami's concept, lines of the geometry are represented by geodesics on the pseudosphere and theorems of non-Euclidean geometry can be proved within ordinary three-dimensional Euclidean space, and not derived in an axiomatic fashion, as Lobachevsky and Bolyai had done previously. In 1840, Ferdinand Minding already considered geodesic triangles on the pseudosphere and remarked that the corresponding "trigonometric formulas" are obtained from the corresponding formulas of spherical trigonometry by replacing the usual trigonometric functions with hyperbolic functions; this was further developed by Delfino Codazzi in 1857, but apparently neither of them noticed the association with Lobachevsky's work.
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William H. Johnson (died 3 March 1883) was a British surveyor in the Great Trigonometric Survey of India. He is noted for the first definition of the eastern boundary of Ladakh along Aksai Chin in the princely state of Jammu and Kashmir, which has come to be called the 'Johnson Line'. After retiring from the Survey of India, Johnson was appointed as the Governor of Ladakh, in which position he served until his death.
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Knowing the castle was lightly guarded, a local merchant Richard Dawkes accompanied by 10 men scaled the cliffs and attacked the porter's lodge, obtaining the keys and entering the castle before the garrison was summoned. Dover Castle was a crucial observation point for the cross-channel sightings of the Anglo-French Survey, which used trigonometric calculations to link the Royal Greenwich Observatory with the Paris Observatory. This work was overseen by General William Roy.
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Exact solutions, based on three-dimensional geometry and spherical trigonometry, began to appear in the mid-9th century. Habash al-Hasib wrote an early example, using an orthographic projection. Another group of solutions uses trigonometric formulas, for example Al-Nayrizi's four-step application of Menelaus's theorem. Subsequent scholars, including Ibn Yunus, Abu al-Wafa, Ibn al-Haitham and Al-Biruni, proposed other methods which are confirmed to be accurate from the viewpoint of modern astronomy.
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The Mathematical Tables Project was one of the largest and most sophisticated computing organizations that operated prior to the invention of the digital electronic computer. Begun in the United States in 1938 as a project of the Works Progress Administration (WPA), it employed 450 unemployed clerks to tabulate higher mathematical functions, such as exponential functions, logarithms, and trigonometric functions. These tables were eventually published in a 28 volume set by Columbia University Press.
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George Joachim Rheticus, most commonly known as Rheticus, was well known for his trigonometric tables and considered a pupil of Copernicus. He was born on February 16, 1514 in Feldkirch, in present day as Austria. After his father’s execution, Rheticus went on to study at the Latin school in Feldkirch, then went to Zurich where he attended the Frauenmuensterschule from 1528 to 1531. In 1533, he began his studies at the University of Wittenberg.
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The TI-83 series is a series of graphing calculators manufactured by Texas Instruments. The original TI-83 is itself an upgraded version of the TI-82. Released in 1996, it was one of the most popular graphing calculators for students. In addition to the functions present on normal scientific calculators, the TI-83 includes many features, including function graphing, polar/parametric/sequence graphing modes, statistics, trigonometric, and algebraic functions, along with many useful applications.
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While this method is attributed to a 1965 paper by James Cooley and John Tukey, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier on the subject in 1807. Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again".
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He invented the calculus of variations including its best-known result, the Euler–Lagrange equation. Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions.
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George Frankland (1800 - 30 December 1838) was an English surveyor and Surveyor-General of Van Diemen's Land (now Tasmania). In 1823, Frankland was appointed surveyor-general at Poona, India, where he became acquainted with Edward Dumaresq. In 1827 Frankland arrived in Van Diemen's Land as first assistant surveyor, in March 1828 he became Surveyor General of Tasmania. Frankland soon began a trigonometric survey of the island, but suffered some criticism due to his slow progress.
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215 These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.Gingerich, Owen. "Islamic astronomy." Scientific American 254.4 (1986): 74-83 The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.
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Goodwin was also the listed author of another program booklet containing tables of trigonometric functions."Seven Place Cosines, Sines, and Tangents For Every Tenth Microturn", Norton Goodwin, Director, Independent Tracking Coordination Program. Reprinted 1964 for the Society of Photographic Scientists and Engineers by H. B. Roebuck and Son, Inc., Baltimore, MD An unusual feature of these tables was the specification of angles in "turns", one turn being 360 degrees, or 2pi radians.
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Ross 640 is a white dwarf star in the northern constellation of Hercules, positioned near the constellation border with Corona Borealis. With an apparent visual magnitude of 13.83, it is too faint to be visible to the naked eye. Its trigonometric parallax from the Gaia mission is , corresponding to a distance of . This compact star has a stellar classification of DZA5.5, indicating a metal-rich atmosphere accompanied by weaker lines of hydrogen.
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In mathematics, the Regiomontanus's angle maximization problem, is a famous optimization problemHeinrich Dörrie,100 Great Problems of Elementary Mathematics: Their History And Solution, Dover, 1965, pp. 369–370 posed by the 15th-century German mathematician Johannes MüllerEli Maor, Trigonometric Delights, Princeton University Press, 2002, pages 46-48 (also known as Regiomontanus). The problem is as follows: The two dots at eye level are possible locations of the viewer's eye. : A painting hangs from a wall.
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Helmut Maier (born 17 October 1953) is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matrix method as well as Maier's theorem for primes in short intervals. He has also done important work in exponential sums and trigonometric sums over special sets of integers and the Riemann zeta function.
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Tracing the Arc is about the Great Trigonometric Survey or Great Arc Project that started in 1802 and lasted through most of the 19th century. Although arising out of cartographic and military necessity, it was an ambitious attempt to measure the curvature of the earth's surface. It remains a major achievement of applied science in British India. The film attempts to recreate the stupendous effort and look at some of its implications.
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The subroutine concept led to the availability of a substantial subroutine library. By 1951, 87 subroutines in the following categories were available for general use: floating point arithmetic; arithmetic operations on complex numbers; checking; division; exponentiation; routines relating to functions; differential equations; special functions; power series; logarithms; miscellaneous; print and layout; quadrature; read (input); nth root; trigonometric functions; counting operations (simulating repeat until loops, while loops and for loops); vectors; and matrices.
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In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in signal processing (to describe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger. The most general identity is given by:Colton & Kress (1998) p. 32.
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The 1865 Trigonometric Survey shows in simple outline two identical-sized buildings with the narrow alleyway leading to unevenly sized rear yards. Another large building fronts Nurses Walk. An annotated drawing on the same base map shows a sanitation line running down the alleyway from about the centre of No. 119 to join with a sewer line running southwards along George Street which discharges as the Queens Wharf Sewer. This sewer was present in 1857.
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Euler invented the calculus of variations including its most well-known result, the Euler–Lagrange equation. Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions.
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She lectured at the Moscow Forestry Institute, the Moscow Polytechnic Institute, and the Sverdlov Communist Institute. Bari applied for and received the only paid research fellowship awarded by the newly created Research Institute of Mathematics and Mechanics. As a student, Bari was drawn to an elite group nicknamed the Luzitania—an informal academic and social organization. She studied trigonometric series and functions under the tutelage of Nikolai Luzin, becoming one of his star students.
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This makes the machinery of complex analysis available. #The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). #Algebraic operations can be done readily on the power series representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.
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He provided accurate trigonometric tables and expressed the theorem in a form suitable for modern usage. As of the 1990s, in France, the law of cosines is still referred to as the Théorème d'Al-Kashi. The theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form.
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This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.
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A page from Ming Antu's Geyuan Milü Jiefa Ming Antu's geometrical model for trigonometric infinite series Ming Antu discovered Catalan numbers Minggatu (Mongolian script: ; , c.1692-c. 1763), full name Sharavyn Myangat () was a Mongolian astronomer, mathematician, and topographic scientist at the Qing court. His courtesy name was Jing An (静安). Minggatu was born in Plain White Banner (now Plain and Bordered White Banner, Xilin Gol League, Inner Mongolia) of the Qing Empire.
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CORDIC uses simple shift-add operations for several computing tasks such as the calculation of trigonometric, hyperbolic and logarithmic functions, real and complex multiplications, division, square-root calculation, solution of linear systems, eigenvalue estimation, singular value decomposition, QR factorization and many others. As a consequence, CORDIC has been used for applications in diverse areas such as signal and image processing, communication systems, robotics and 3D graphics apart from general scientific and technical computation.
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Communication mast and trigonometric point on Frewin Hill in Portloman parish Portloman is one of 8 civil parishes in the barony of Corkaree in the Province of Leinster. The civil parish covers . Portloman civil parish comprises 8 townlands: Ballard, Ballyboy, Balrath, Grangegeeth, Monroe, Portloman, Scurlockstown and Wattstown. The neighbouring civil parishes are: Portnashangan to the north, Rathconnell (barony of Moyashel and Magheradernon) to the east, Templeoran (barony of Moygoish) to the south and west.
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1900 BC), some have even asserted that the ancient Babylonians had a table of secants.Joseph (2000b, pp.383-84). There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table. The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c.
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A line through P (except the vertical line) is determined by its slope. Furthermore, each of the lines (except the vertical line) intersects the unit circle in exactly two points, one of which is P. This determines a function from points on the unit circle to slopes. The trigonometric functions determine a function from angles to points on the unit circle, and by combining these two functions we have a function from angles to slopes.
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The 13 chapters of the second part cover the nature of the sphere, as well as significant astronomical and trigonometric calculations based on it. Nilakantha Somayaji's astronomical treatise the Tantrasangraha similar in nature to the Tychonic system proposed by Tycho Brahe had been the most accurate astronomical model until the time of Johannes Kepler in the 17th century.George G. Joseph (2000). The Crest of the Peacock: Non-European Roots of Mathematics, p. 408.
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The requirement for accurate maps in Australia led to the development of a control network of Trig stations. The word "trig" is short for trigonometric. Trig stations can be located on landmarks such as hills or high man-made structures such as buildings, or grain silos or water reservoirs. The process of triangulation was used as the method to provide the locations of the trig stations, which were then used to connect to smaller scale surveys or for mapping.
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Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg and Dresden, he entered the Polytechnische Schule, now Technische Universität, in Dresden to study engineering science in 1859. Finding him especially enthusiastic about geodesy, one of his teachers, Christian August Nagel, hired him while still a student to work on the triangulation of the Erzgebirge and the drafting of the trigonometric network for Saxony. In 1863 Helmert became Nagel's assistant on the Central European Arc Measurement.
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In classical potential theory, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.
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KCalc is the software calculator integrated with the KDE Software Compilation. In the default view it includes a number pad, buttons for adding, subtracting, multiplying, and dividing, brackets, memory keys, percent, reciprocal, factorial, square, and x to the power of y buttons. Additional buttons for scientific and engineering (trigonometric and logarithmic functions), statistics and logic functions can be enabled as needed. 6 additional buttons can be predefined with mathematical constants and physical constants or custom values.
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The 9100A was the first scientific calculator by the modern definition, i.e., capable of trigonometric, logarithmic (log/ln), and exponential functions, and was the beginning of Hewlett-Packard's long history of using Reverse Polish notation (RPN) entry on their calculators. Due to the similarities of the machines, Hewlett-Packard was ordered to pay about $900,000 in royalties to Olivetti after copying some of the solutions adopted in the Programma 101, like the magnetic card and the architecture.
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Mean-field-type filters, which are filters that depend on the distribution of the state, were first proposed by L&G; Lab members. They provided explicit solutions to a class of mean-field-type games with non- linear state dynamics and or non-quadratic cost functions. The non-linearity includes trigonometric functions, hyperbolic functions, logarithmic functions and power (polynomial) cost functions. Tembine has worked on game theory with small, medium and large number of interacting agents.
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In 1815, Peter Mark Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers. In 1821, Nathaniel Bowditch, described in the American Practical Navigator a "sliding rule" that contained scales trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.
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Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itself came later); there is a story from Anthony à Wood, perhaps not well substantiated, that Napier had a hint from Craig that Longomontanus, a follower of Brahe, was working in a similar direction. It has been shown that Craig had notes on a method of Paul Wittich that used trigonometric identities to reduce a multiplication formula for the sine function to additions.
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However, one of his papers, on uniform convergence of trigonometric series, remains well cited.. 49 citations in Google scholar as of 2011-03-30. He was also a contributor to the Encyclopædia Britannica. On 1 July 1899 at St Paul's Church, Oxford he married Eliza Ostler of 14 Walton Crescent, the daughter of the tailor James Ostler, and they had three children. Jolliffe died on 17 March 1944 in Oxford, and was buried in St Sepulchre's Cemetery, Walton Street.
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XNUMBERS is a multi-precision floating point computing and numerical methods library for Microsoft Excel. Xnumbers claims to be an open source Excel addin (xla), the license however is not an open source license. XNUMBERS performs multi-precision floating point arithmetic from 15 up to 200 significant digits. The version 5.6 as of 2008 is compatible with Excel 2003/XP and consists of a set of more than 300 functions for arithmetic, complex, trigonometric, logarithm, exponential calculus.
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It is possible to describe the five angles of any convex equilateral pentagon with only two angles α and β, provided α ≥ β and δ is the smallest of the other angles. Thus the general equilateral pentagon can be regarded as a bivariate function f(α, β) where the rest of the angles can be obtained by using trigonometric relations. The equilateral pentagon described in this manner will be unique up to a rotation in the plane.
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The initialization of the error term is derived from an offset of ½ pixel at the start. Until the intersection with the perpendicular line, this leads to an accumulated value of r in the error term, so that this value is used for initialization. The frequent computations of squares in the circle equation, trigonometric expressions and square roots can again be avoided by dissolving everything into single steps and using recursive computation of the quadratic terms from the preceding iterations.
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The choice above, however, is typical because it has two useful properties. First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited. Second, if the x_n are real numbers, then p(t) is real as well. In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to N-1 (instead of roughly -N/2 to +N/2 as above), similar to the inverse DFT formula.
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A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss.
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Each trigonometric and hyperbolic function has its own name and abbreviation both for the reciprocal (for example, ), and its inverse (for example ). A similar convention exists for logarithms, where today usually means , not . To avoid ambiguity, some mathematicians choose to use to denote the compositional meaning, writing for the -th iterate of the function , as in, for example, meaning . For the same purpose, was used by Benjamin Peirce whereas Alfred Pringsheim and Jules Molk suggested instead.
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Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography, given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Multi Party Computation, Secret Sharing schemes.
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In mathematics, an elementary function is a function of a single variable composed of particular simple functions. Elementary functions are typically defined as a sum, product, and/or composition of finitely many polynomials, rational functions, trigonometric and exponential functions, and their inverse functions (including arcsin, log, x1/n). Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.... An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s..
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JSBSim uses a coefficient build-up method for modeling the aerodynamic characteristics of aircraft. Any number of forces and moments (or none at all) can be defined for each of the axes. Each force/moment specification includes a definition comment, and a specification of the function that calculates the force or moment. The function definition can be a simple value, or a complicated function that includes trigonometric and logarithmic functions, and a one-, two-, or three-dimensional table lookup.
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In order to work out the masses of the components of a visual binary system, the distance to the system must first be determined, since from this astronomers can estimate the period of revolution and the separation between the two stars. The trigonometric parallax provides a direct method of calculating a star's mass. This will not apply to the visual binary systems, but it does form the basis of an indirect method called the dynamical parallax.
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The human naked eye has an angular resolution of approximately 280 microradiansMiller, David; Schor, Paulo; Peter Magnante. "Optics of the Normal Eye", pg. 54 of Ophthalmology by Yanoff, Myron; Duker, Jay S. (μrad) (approx 0.016° or 1 minute of arc), and the ISS targets an altitude of 400 km. Using basic trigonometric relations, this means that an astronaut on the ISS with 20/20 vision could potentially detect objects that are 112 m or greater in all dimensions.
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However, since this would be at the absolute limit of the resolution, objects on the order of 100 m would appear as unidentifiable specks, if not rendered invisible due to other factors, such as atmospheric conditions or poor contrast. For readability from the ISS, using the same trigonometric principles and a recommended character size of about 18 arcminutes, or about 5,000 μrad, each letter would need to be about 2 km tall for clear legibility in good conditions.
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Mount Sanford was exhumed to enlarge the park boundaries in 1996. The Wickham flows for only six months of the year, during the wet season. During the dry season, a string of waterholes form along the course of the river some of which are spring-fed and almost permanent. When Lawrence Wells started his trigonometric survey of the Northern Territory in 1905 he formed his depot on the Wickham River not too far from Victoria River Downs Station homestead.
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In 1619 Speidell published a table entitled "New Logarithmes" in which he calculated the natural logarithms of sines, tangents, and secants. He then diverged from Napier's methods in order to ensure all of the logarithms were positive. A new edition of "New Logarithmes" was published in 1622 and contained an appendix with the natural logarithms of all numbers 1-1000. Along with William Oughtred and Richard Norwood, Speidell helped push toward the abbreviations of trigonometric functions.
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ROM "B" of the Galaksija is a 2732 EPROM chip that contains extensions to the original Galaksija BASIC available in base ROM ("A"). It was labeled as "B" by the creator of the Galaksija, Voja Antonić, but was commonly referred to as "ROM 2". ROM "B" contained added Galaksija BASIC commands and functions (mostly trigonometric) as well as a Z80 assembler and a machine code monitor. This ROM was not required and was an optional upgrade.
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The DSP-1 is the most varied and widely used of the SNES DSPs, appearing in over 15 separate titles. It is used as a math coprocessor in games such as Super Mario Kart and Pilotwings that require more advanced Mode 7 scaling and rotation. It also provides fast support for the floating point and trigonometric calculations needed by 3D math algorithms. The later DSP-1A and DSP-1B serve the same purpose as the DSP-1.
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In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically, usually by series expansions. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects.Bate, Mueller, White (1971): p.
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56 and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000 Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2 times the height, or h:b = 4:. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler triangle.
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In his work on heat conduction, Joseph Fourier maintained that the arbitrary function may be represented as an infinite trigonometric series and made explicit the possibility of expanding functions in terms of Bessel functions and Legendre polynomials, depending on the context of the problem. It took some time for his ideas to be accepted as his use of mathematics was less than rigorous. Although initially skeptical, Poisson adopted Fourier's method. From around 1815 he studied various problems in heat conduction.
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In recognition of Sikdar's mathematical genius, the German Philosophical Society made him a Corresponding Member in 1864, a very rare honour.See Deepak Kumar, Patterns of Colonial Science in India, Indian Journal of History of Science, 15(1), May 1980. The Department of Posts, Government of India, launched a postal stamp on 27 June 2004, commemorating the establishment of the Great Trigonometric Survey in Chennai, India on 10 April 1802. The stamps feature Radhanath Sikdar and Nain Singh, two significant contributors to society.
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Much like in The Incredible Machine, users can solve a variety of puzzles using a limited selection of parts or tinker with the freeform mode. Widget Workshop focuses more on the freeform mode than the other game. Unlike the Rube Goldberg nature of The Incredible Machine, the parts in Widget Workshop are not restricted to the mechanical or physical. Items include display boxes, graphing windows, random number generators, and mathematical tools ranging from addition and subtraction to Boolean logic gates and trigonometric functions.
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Starting with known special cases, the calculation of logarithms and trigonometric functions can be performed by looking up numbers in a mathematical table, and interpolating between known cases. For small enough differences, this linear operation was accurate enough for use in navigation and astronomy in the Age of Exploration. The uses of interpolation have thrived in the past 500 years: by the twentieth century Leslie Comrie and W.J. Eckert systematized the use of interpolation in tables of numbers for punch card calculation.
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Erdélyi started his career studying Markov and Bernstein inequalities for constrained polynomials in the late eighties. In his Ph.D. dissertation he extended many important polynomial inequalities to generalized polynomials by writing the generalized degree in place of the ordinary. His trigonometric work on Remez inequality represents one of his most cited papers. In 1995, he finished his Springer-Verlag graduate text Polynomials and Polynomial Inequalities, co-authored with Peter Borwein, and including an appendix proving the irrationality of ζ(2) and ζ(3).
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Also they presented a near-infrared (J-band) trigonometric parallax of the system, measured using WIRCam on the Canada-France-Hawaii Telescope (CFHT), Mauna Kea, in seven epochs during the 2009–2010; and spectroscopy with the X-Shooter spectrograph at the European Southern Observatory's Very Large Telescope (VLT) Unit Telescope 2 (UT2) in Chile (the observations have been performed from May 5 to July 9, 2010), that allowed to calculate the temperature (and other physical parameters) of the two brown dwarfs.
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Oldham earned a B.A. degree from Boston University and an M.A. from Allegheny College, and became a government surveyor. He was handpicked for the Great Trigonometric Survey of India, a key 19th century survey of India and its adjoining lands. It was in 1873, in the midst of this secular work, that Oldham was invited to the preaching tents of visiting American Daniel O. Fox. The teachings of these Methodist missionaries, led by Bishop William Taylor, were strange yet attractive to Oldham.
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Such a design leads to a matrix: columns represent increments in calculator functionality, and rows represent different presentation front-ends. Such a matrix M is shown to the right: columns allow one to pair basic calculator functionality (base) with optional logarithmic/exponentiation (lx) and trigonometric (td) features. Rows allow one to pair core functionality with no front-end (core), with optional GUI (gui) and web-based (web) front-ends. An element Mij implements the interaction of column feature i and row feature j.
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In the Circles of Proportion and the Horizontal Instrument, Oughtred introduces the abbreviations for trigonometric functions. This book was originally in manuscript before it eventually became published. Also, the slide rule is discussed, an invention that was made by Oughtred which provided a mechanical method of finding logarithmic results. It is mentioned in this book that John Napier was the first person to ever use to the decimal point and comma, however Bartholomaeus Pitiscus was actually the first to do so.
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Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals and to giveJ. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", American Mathematical Monthly, 46 (1939) 345–347. (JSTOR)E. W. Hobson: A Treatise on Plane Trigonometry.
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If a function of the form y=f(x) cannot be postulated, one can still try to fit a plane curve. Other types of curves, such as conic sections (circular, elliptical, parabolic, and hyperbolic arcs) or trigonometric functions (such as sine and cosine), may also be used, in certain cases. For example, trajectories of objects under the influence of gravity follow a parabolic path, when air resistance is ignored. Hence, matching trajectory data points to a parabolic curve would make sense.
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The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must not encapsulate the specification of the anglesum being 180 degrees.Hyperbolic Barycentric Coordinates, Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp.
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The original description of the phenomenon was presented in a paper by Thomas E. Lutz and Douglas H. Kelker in the Publications of the Astronomical Society of the Pacific, Vol. 85, No. 507, p. 573 article entitled "On the Use of Trigonometric Parallaxes for the Calibration of Luminosity Systems: Theory." although it was known following the work of Trumpler & Weaver in 1953. The discussion on statistical bias on measurements in astronomy date back to as early as to Eddington in 1913.
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Automorphic functions then generalize both trigonometric and elliptic functions. Poincaré explains how he discovered Fuchsian functions: :For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep.
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Pre-Calculus served as the introductory mathematics class at TGA for those who had not taken the course elsewhere or those who are not prepared to take Calculus upon arrival at TGA. The course was similar in content to the Math 130 course offered at the University of Tennessee at Knoxville. Generally speaking, the course content serves to prepare juniors to take Calculus during the Spring semester of the junior year. Content includes a review of algebraic, logarithmic, exponential, and trigonometric functions.
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Therefore, one uses the radian as angular unit: a radian is the angle that delimits an arc of length on the unit circle. A complete turn is thus an angle of radians. A great advantage of radians is that they make many formulas much simpler to state, typically all formulas relative to derivatives and integrals. Because of that, it is often understood that when the angular unit is not explicitly specified, the arguments of trigonometric functions are always expressed in radians.
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In fact, it can be traced all the way back to the Hellenistic Civilization. While people have devised such machines over the centuries, mathematicians continued to perform calculations by hand, a machines offered little advantage in speed. For complicated calculations, they employed tables, especially of logarithmic and trigonometric functions, which were computed by hand. But right in the middle of the Industrial Revolution in England, Charles Babbage thought of using the all-important steam engine to power a mechanical computer, the Difference Engine.
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In computer programming languages, the inverse trigonometric functions are usually called by the abbreviated forms asin, acos, atan. The notations , , , etc., as introduced by John Herschel in 1813, are often used as well in English-language sources—conventions consistent with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions such as , which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse or reciprocal and compositional inverse.
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65 Ursae Majoris, abbreviated as 65 UMa, is a star system in the constellation of Ursa Major. With an apparent magnitude of about 6.5, it is at the limit of human eyesight and is just barely visible to the naked eye. Trigonometric parallax measurements made by the Hipparcos spacecraft put it at a distance of about 690 light years (210 parsecs); this is in close agreement with the dynamical parallax value of . 65 Ursae Majoris is a sextuple star system.
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An implied assumption for the above equation to be true is that r_1 and r_2 relate to the same position of P. When P is a vehicle, then typically r_1 and r_2 must be measured within a synchronization tolerance that depends on the vehicle speed and the allowable vehicle position error. Alternatively, vehicle motion between range measurements may be accounted for, often by dead reckoning. A trigonometric solution is also possible (side-side- side case). Also, a solution employing graphics is possible.
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Epsilon Trianguli, Latinized from ε Trianguli, is a binary star system in the northern constellation of Triangulum. Based upon measurement of its trigonometric parallax, it is approximately 390 light years from Earth. The primary component is an A-type main sequence star with a stellar classification of A2 V, an apparent magnitude of +5.50 and an estimated age of 600 million years. It has 2.75 times the mass of the Sun and is spinning with a projected rotational velocity of 107 km/s.
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Mahematical concepts he uses in his work include trigonometric functions, exponential function, Fibonacci sequence, sawtooth wave, etc. His artwork 9,000 Ellipses was used as the background cover image of The American Mathematical Monthly – November 2017. His artwork Heart was used as the image for the February page of the 2019 Calendar of Mathematical Imagery published by the American Mathematical Society. His artwork Bird was used as the postcard image of the Art ∩ Math exhibit held at Center on Contemporary Art, Seattle in 2018.
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He was also a pioneer in spherical trigonometry. In 830 AD, Habash al-Hasib al-Marwazi produced the first table of cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. By the 10th century AD, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions.Boyer (1991) p. 238.
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It had a complete instruction set for controlling its arithmetic units. The algorithms for trigonometric and logarithmic algorithms took advantage of this instruction set which has put the slide rule out of business. But it also opened the door to a new genre of pocket calculators. Using the same hardware as the HP-35, France designed the HP-80 business calculator, that replaced reams of tables used to compute mortgages, returns on investments and other business transactions with a dedicated keyboard.
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A chart to convert between degrees and radians In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2' radians, so 180° is equal to radians, or equivalently, the degree is a mathematical constant: 1° = .
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The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities. Unless otherwise specified, this article deals with triangles in the Euclidean plane.
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Sangamagrama Mādhava was a prominent Kerala mathematician-astronomer who founded the Kerala school of astronomy and mathematics. Saṅgamagrāma in medieval Kerala believed to be Irinjalakuda Brahminical grama, including Aloor. Sangamagrama Mādhava is described as "the greatest mathematician-astronomer of medieval India" or as "the founder of mathematical analysis". He was the first to have developed infinite series of approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".
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Late in life, when he was 57, Roy was granted the opportunity to establish his lasting reputation in the world of geodesy. The opening came from a completely unexpected direction. In 1783 Cassini de Thury addressed a memoir to the Royal Society in which he expressed grave reservations of the measurements of latitude and longitude which had been undertaken at Greenwich Observatory. He suggested that the correct values might be found by combining the Paris Observatory figures with a precise trigonometric survey between the two observatories.
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In mathematics, a transcendental curve is a curve that is not an algebraic curve.Newman, JA, The Universal Encyclopedia of Mathematics, Pan Reference Books, 1976, , "Transcendental curves". Here for a curve, C, what matters is the point set (typically in the plane) underlying C, not a given parametrisation. For example, the unit circle is an algebraic curve (pedantically, the real points of such a curve); the usual parametrisation by trigonometric functions may involve those transcendental functions, but certainly the unit circle is defined by a polynomial equation.
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In 1915, the Scottish astronomer Robert Innes, Director of the Union Observatory in Johannesburg, South Africa, discovered a star that had the same proper motion as Alpha Centauri. This is the original Proxima Centauri discovery paper. He suggested that it be named Proxima Centauri (actually Proxima Centaurus). In 1917, at the Royal Observatory at the Cape of Good Hope, the Dutch astronomer Joan Voûte measured the star's trigonometric parallax at and determined that Proxima Centauri was approximately the same distance from the Sun as Alpha Centauri.
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He developed trigonometry and constructed trigonometric tables, and he solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. His other reputed achievements include the discovery and measurement of Earth's precession, the compilation of the first comprehensive star catalog of the western world, and possibly the invention of the astrolabe, also of the armillary sphere, which he used during the creation of much of the star catalogue.
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The algorithms used by NOVAS are based on vector astrometry theories and the IAU resolutions. Instead of using trigonometric formulae from spherical astrometry, NOVAS uses the matrix and vector formulation which is more rigorous. This version implements the resolutions on astronomical reference systems and Earth rotation models passed at the IAU General Assemblies in 1997, 2000, and 2006. According to the Astronomical Applications Department, the algorithms used in NOVAS are identical to those used in the production of the US part of the Astronomical Almanac.
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The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory.
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These formulas used infinite series and trigonometric functions to calculate pi to hundreds of decimal places. Computers were used in the 20th century to calculate pi and its value was known to one billion decimals places by 1989. One reason to accurately calculate pi is to test the performance of computers. Another reason is to determine if pi is a specific fraction, which is a ratio of two integers called a rational number that has a repeating pattern of digits when expressed in decimal form.
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Casio BASIC is a programming language used in the Casio calculators such as the Classpad, PRIZM Series, fx-9860G Series, fx-5800P, Algebra FX and CFX graphing calculators. The language is a linear structured, BASIC-based programming language. It was devised to allow users to program in commonly performed calculations, such as the Pythagorean theorem and complex trigonometric calculations. Output from the program can be in the form of scrolling or located text, graphs, or by writing data to lists in the calculator memory.
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On the back of the mater, there is often engraved a number of scales that are useful in the astrolabe's various applications. These vary from designer to designer, but might include curves for time conversions, a calendar for converting the day of the month to the sun's position on the ecliptic, trigonometric scales, and graduation of 360 degrees around the back edge. The alidade is attached to the back face. An alidade can be seen in the lower right illustration of the Persian astrolabe above.
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In about 1970 HP co-founder Bill Hewlett challenged his co-workers to create a "shirt-pocket sized HP-9100". At the time, slide rules were the only practical portable devices for performing trigonometric and exponential functions, as existing pocket calculators could only perform addition, subtraction, multiplication, and division. Introduced at $395 (),This is a best practices guess of the inflation adjusted price. like HP's first scientific calculator, the desktop 9100A, it used Reverse Polish Notation (RPN) rather than what came to be called "algebraic" entry.
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The reduction formula can be derived using any of the common methods of integration, like integration by substitution, integration by parts, integration by trigonometric substitution, integration by partial fractions, etc. The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by In, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example In-1 or In-2. This makes the reduction formula a type of recurrence relation.
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Although it was originally published in 1550, it was subsequently updated 3 times in 1554, 1579, and 1587. The original 1550 edition incorporated figures explaining the measurement of spheres and the determination of geographical coordinates. The 1554 edition adds to this by providing the calculations necessary for discerning the distance between two points from coordinates. The works found in this book are based on the trigonometric tables of Copernicus, the flat and spherical geometry developments of Georg Joachim Rheticus, and works of Johnannes Regiomantus.
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HP also sold a number of program collections for scientific and engineering applications on sets of pre-recorded (and write-protected) cards. The HP-65 had a "feature" whereby storage register R9 was corrupted whenever the user (or program) executed trigonometric functions or performed comparison tests; this kind of issue was common in many early calculators, caused by a lack of memory due to cost, power, and/or size considerations. Since the limitation was documented in the manual, it is not strictly speaking a bug.
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Before the Moreton Bay penal settlement, Mount Coot-tha was the home of the Yugara Aboriginal people.Mount Coot-tha Forest track map — (Brisbane City Council) The Aboriginal people came to the area to collect 'ku-ta' (honey) that was produced by the native stingless bee. In 1839, surveyor James Warner and his team cleared the top of the mountain of all trees except one large eucalypt tree. Because this tree could be seen from many other locations, they used as a trigonometric station to take surveying measurements.
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The distance to Beta Pictoris and many other stars was measured by the Hipparcos satellite. This was done by measuring its trigonometric parallax: the slight displacement in its position observed as the Earth moves around the Sun. Beta Pictoris was found to exhibit a parallax of 51.87 milliarcseconds, a value which was later revised to 51.44 milliarcseconds when the data was reanalyzed taking systematic errors more carefully into account. The distance to Beta Pictoris is therefore 63.4 light years, with an uncertainty of 0.1 light years.
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The trig station was '89 feet high' and constructed from timber carried up by manual labour from the valley below. The trigonometric station was called 'The Stockade' by locals, on account of the palisade surrounding the central cairn, but by the turn of the 20th century it had largely disappeared. A full survey of Ben Lomond was conducted from September 1905 to 1912 by Colonel William Vincent Legge, Stacks Bluff was found to be the second highest feature on the plateau at this time.
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Before Legges Tor was surveyed, Stacks Bluff (at the plateau's southern extremity) was thought to be the highest elevation on the Ben Lomond plateau. From 1905 to 1912 a full survey of Ben Lomond was conducted by Legge and his survey party. The survey party explored the highlands on the north of the plateau in 1907. Legge had long suspected that the north of the plateau was higher than the trigonometric station on Stacks Bluff although it is less obviously elevated from casual observation.
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PROIV has little or no object-orientation and makes only a limited amount of procedural abstraction available to the programmer. PROIV has little support for analytical/statistical/mathematical functions; for example, it does not include basic trigonometric functions. The PROIV-supplied "GUI client", which renders the rich-client UI for applications written in PROIV, is based around ActiveX technology and works only on Windows client platforms. Consequently, the programmers' development environments supplied with more-recent PROIV releases also work only on a Windows client platform.
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Bald Hill was to become the primary reference point for surveying the region. In 1836, the surveyor Thomas Watson returned to area and used Bald Hill as the principal trigonometric reference. Watson was to map out a number of lots in the area including the western boundary for Beverley town site. Two of the lots surveyed were Avon location 14 with , and Avon location K with ; location 14 was given to Captain Mark Currie, Fremantle Harbour Master, while location K was given to Stirling.
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AD Leonis is an M-type star with a spectral type M3.5eV, indicating it is a main sequence star that displays emission lines in its spectrum. At a trigonometric distance of , it has an apparent visual magnitude of 9.43. It has about 39–42% of the Sun's mass — above the mass at which a star is fully convective — and 39% of the Sun's radius. The projected rotation of this star is only 3 km/s, but it completes a rotation once every 2.24 days.
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One of the greatest projects of 19th century geography was the Great Trigonometric Survey of India. The British also wanted geographical information on the lands further north. This was not just out of scientific curiosity: The Russians were attempting to expand their empire into Central Asia, and the British feared that they might have set their eyes on gaining the riches of India, which was at that time a British colony. Thus, the Russians and the British both tried to extend their influence in Asia.
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Minkowski's space-time formalism was quickly accepted and further developed. For example, Arnold Sommerfeld (1910) replaced Minkowski's matrix notation by an elegant vector notation and coined the terms "four vector" and "six vector". He also introduced a trigonometric formulation of the relativistic velocity addition rule, which according to Sommerfeld, removes much of the strangeness of that concept. Other important contributions were made by Laue (1911, 1913), who used the spacetime formalism to create a relativistic theory of deformable bodies and an elementary particle theory.
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She presented the main result of her research to the Moscow Mathematical Society in 1922—the first woman to address the society. In 1926, Bari completed her doctoral work on the topic of trigonometric expansions, winning the Glavnauk Prize for her thesis work. In 1927, Bari took advantage of an opportunity to study in Paris at the Sorbonne and the College de France. She then attended the Polish Mathematical Congress in Lwów, Poland; a Rockefeller grant enabled her to return to Paris to continue her studies.
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The IEEE Spectrum has organized a number of competitions on quilt block design, and several books have been published on the subject. Notable quiltmakers include Diana Venters and Elaine Ellison, who have written a book on the subject Mathematical Quilts: No Sewing Required. Examples of mathematical ideas used in the book as the basis of a quilt include the golden rectangle, conic sections, Leonardo da Vinci's Claw, the Koch curve, the Clifford torus, San Gaku, Mascheroni's cardioid, Pythagorean triples, spidrons, and the six trigonometric functions..
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The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both Euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must not encapsulate the specification of the angle sum being 180 degrees.Hyperbolic Barycentric Coordinates, Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp.
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Von Neumann describes a detailed design of a "very high speed automatic digital computing system." He divides it into six major subdivisions: a central arithmetic part, CA, a central control part, CC, memory, M, input, I, output, O, and (slow) external memory, R, such as punched cards, Teletype tape, or magnetic wire or steel tape. The CA will perform addition, subtraction, multiplication, division and square root. Other mathematical operations, such as logarithms and trigonometric functions are to be done with table look up and interpolation, possibly biquadratic.
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He was born at Bristol. At first he intended to study law; but he gave up the idea on his father's death in 1794. He entered the army, obtaining a commission in the 12th Regiment of Foot, then stationed in India, where he assisted William Lambton in the Great Trigonometric Survey. Failing health obliged him to return to England; and in 1808, then a lieutenant, he entered on a student career at the Senior Division of the new Royal Military College at High Wycombe.
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It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method. In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.
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Seleucus may have proved the heliocentric theory by determining the constants of a geometric model for the heliocentric theory and developing methods to compute planetary positions using this model. He may have used early trigonometric methods that were available in his time, as he was a contemporary of Hipparchus. A fragment of a work by Seleucus has survived in Arabic translation, which was referred to by Rhazes (b. 865). Alternatively, his explanation may have involved the phenomenon of tides,Lucio Russo, Flussi e riflussi, Feltrinelli, Milano, 2003, .
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Indian people have played a major role in the development of the philosophy, sciences, mathematics, arts, architecture and astronomy throughout history. During the ancient period, notable mathematics accomplishment of India included Hindu–Arabic numeral system with decimal place-value and a symbol for zero, interpolation formula, Fibonacci's identity, theorem, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations. Chakravala method, sign convention, madhava series, and the sine and cosine in trigonometric functions can be traced to the jyā and koti-jyā.Boyer, Carl B. (1991).
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The hill lies in the "Hochsolling", the central and highest part of the Solling, which is surrounded by the Solling-Vogler Nature Park. This heavily wooded hill is a little south of the half way point between Boffzen and Dassel, as the crow flies, and around 1.5 km east of Neuhaus. From topographical maps it is clear, for example from trigonometric points that there are three different summit on the Moosberg at 513.0 m (north), 508.7 m (centre) and 508.6 m (south). On the western slope of the Moosberg is the Hochsolling Observation Tower.
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Two uniformly distributed values, u and v are used to produce the value , which is likewise uniformly distributed. The definitions of the sine and cosine are then applied to the basic form of the Box–Muller transform to avoid using trigonometric functions. The polar form was first proposed by J. Bell and then modified by R. Knop. While several different versions of the polar method have been described, the version of R. Knop will be described here because it is the most widely used, in part due to its inclusion in Numerical Recipes.
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Colonel Sir George Everest CB FRS FRAS FRGS (; 4 July 1790 – 1 December 1866) was a British surveyor and geographer who served as Surveyor General of India from 1830 to 1843. He is best known for having Mount Everest, the highest mountain on Earth, named in his honour. After receiving a military education in Marlow, Everest joined the East India Company and arrived in India at the age of 16. He was eventually made an assistant to William Lambton on the Great Trigonometric Survey, and replaced Lambton as superintendent of the survey in 1823.
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Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered.A. Lipson, S.G. Lipson, H. Lipson, Optical Physics, 4th edition, 2010, University Press, Cambridge, UK, p. 51. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere.
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Concentration of dots gives a glance at the distribution of stress direction. :2) Fault Motion Fault movement is resolved into three components (as in 3D), which are vertical transverse, horizontal transverse and lateral components, by trigonometric relation with the measured dips and trends. Net slip is shown more clearly which paves the way to understanding the deformation. 3) Individual fault geometry represented on a stereonet :3) Individual Fault Geometry Fault planes are represented by lines in stereonets (equal area lower hemisphere projection), while rakes on them are indicated by dots sitting on the lines.
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He published an elegant trigonometric solution of Malfatti's problem in the French math journal Nouvelles Annales de Mathématiques, but due to a copy error the author's name was given as Lechmütz. In 1840 Lehmus wrote a letter to the French mathematician C. Sturm asking him for an elementary geometric proof of the theorem that is now named after him. Sturm passed the problem on to other mathematicians and Jakob Steiner was one of the first who provided a proof. In 1850 Lehmus came up with a different proof on his own.
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Ngga Pulu is a summit on the north rim of Mount Carstensz in the western part of the island of New Guinea rising . Trigonometric measurements showed that Ngga Pulu was (and had been for many centuries before) the highest mountain of New Guinea and also the highest summit of the Australia-New Guinea continent. The elevation of Ngga Pulu in 1936 was about , and it was the highest and most prominent peak between the Himalaya and the Andes. However, due to glacial melting, Ngga Pulu lost a lot of elevation in the 20th century.
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Lieutenant-Colonel Thomas George Montgomerie FRS (1830–1878) was a British surveyor who participated in the Great Trigonometric Survey of India as a lieutenant in the 1850s. He was the person to label K2, the second highest mountain in the world, the K standing for Karakoram. The label "K2" has stuck and has become, and remains, the mountain's most commonly used name. Despite being often denied close range access, the 19th century survey work carried out by Montgomerie and the survey of India has been shown to be accurate.
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In 1831, the Surveyor General of India George Everest was searching for a brilliant young mathematician with a particular proficiency in spherical trigonometry, the math teacher of Indian "Hindu" College John Tytler recommended his pupil Sikdar, then only 19. Sikdar joined the Great Trigonometric Survey in December 1831 as a "computer" at a salary of thirty rupees per month. Soon he was sent to Sironj (near Dehradun) where he excelled in geodetic survey. Apart from mastering the usual geodetic processes, he invented quite a few of his own.
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Scientific mode supports exponents and trigonometric functions, and programmer mode gives the user access to more options related to computer programming. The Calculator program has a long associated history with the beginning of the Macintosh platform, where a simple four-function calculator program was a standard desk accessory from the earliest system versions. Though no higher math capability was included, third-party developers provided upgrades, and Apple released the Graphing Calculator application with the first PowerPC release (7.1.2) of the Mac OS, and it was a standard component through Mac OS 9.
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In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics and later by Gottfried Wilhelm Leibniz, among others. These expressions are the Maclaurin series expansions of the trigonometric sine, cosine and arctangent functions, and the special case of the power series expansion of the arctangent function yielding a formula for computing π.
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From Gyrovector space#triangle center The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must not encapsulate the specification of the anglesum being 180 degrees.Hyperbolic Barycentric Coordinates, Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp.
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Like many of the surrounding hill areas, the Boginderra Hills may have seen light use by Australian Aborigines for various purposes (though evidence has yet been found). Immediately prior to the formation of the reserve, the land had been used for livestock grazing. Cropping was impossible on the rocky and hilly terrain and so no substantial land clearing had taken place on the hill itself (compared to the plains that surround it). A trigonometric station, called Nurraburra, was placed on the hill by the Central Mapping Authority (now part of NSW Land and Property Information).
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This is the case of distributivity or trigonometric identities. For example, the distributivity law allows rewriting (x+1)^4 \rightarrow x^4+4x^3+6x^2+4x+1 and (x-1)(x^4+x^3+x^2+x+1) \rightarrow x^5-1. As there is no way to make a good general choice of applying or not such a rewriting rule, such rewritings are done only when explicitly asked for by the user. For the distributivity, the computer function that applies this rewriting rule is generally called "expand".
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Madhava extended Archimedes' work on the geometric Method of Exhaustion to measure areas and numbers such as π, with arbitrary accuracy and error limits, to an algebraic infinite series with a completely separate error term. This implies that he understood very well the limit nature of the infinite series. Thus, Madhava may have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series, and rational approximations of infinite series. However, as stated above, which results are precisely Madhava's and which are those of his successors is difficult to determine.
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Gravimeter with variant of Repsold pendulum The large increase in gravity measurement accuracy made possible by Kater's pendulum established gravimetry as a regular part of geodesy. To be useful, it was necessary to find the exact location (latitude and longitude) of the 'station' where a gravity measurement was taken, so pendulum measurements became part of surveying. Kater's pendulums were taken on the great historic geodetic surveys of much of the world that were being done during the 19th century. In particular, Kater's pendulums were used in the Great Trigonometric Survey of India.
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In 1832 the Land Survey Department, under Surveyor-General George Frankland, began preparing for a trigonometric survey of the island, and Darke was successful in obtaining a temporary position within the department. One area of particular interest to Frankland was the area west of Wylds Craig (then known as the Peak of Teneriffe). An escaped convict from Sarah Island, James Goodwin, has passed through the area on his way back to the settled districts. With five men, Darke was sent to explore the region west of Wylds Craig beginning on 19 March 1833.
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The pitch of a roof is its vertical 'rise' over its horizontal 'span'. However, most often a ratio of "pitch" (also fraction) is slang used for the (more useful) 'slope' (of rise over 'run') of just one side (half the span) of a dual pitched roof. This is the 'slope' of geometry, stairways and other construction disciplines, or the trigonometric arctangent function of its decimal fraction. In the imperial measurement systems, "pitch" is usually expressed with the rise first and run second (in the USA, run is held to number 12).
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Forward vs. Inverse Kinematics In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics.
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The named Wormberch had already surfaced by the 13th century in the documents and commodity schedules of the County of Regenstein-Blankenburg in connexion with the mining of iron ore. In the 19th century the mountain was still being called Wormsberg or Wormberg, but a convincing derivation of the name has yet to be found. Around 1850 the first trig post was erected on the summit of the Wurmberg to assist in surveying the Harz mountains. It was replaced in 1890 by a wooden tower, also used for trigonometric measurements, that stood until 1930.
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The ruins of this hut and the stone cairn mentioned by Pröhle, were used in 1890 for building the above-mentioned trigonometric tower. The circular site first appeared during the construction of this tower as an abutment for the diagonal posts that supported the tower on all sides. And on one of the stones of the Hexentreppe, an English button from the period around 1800 was found, which finally proved the staircase to be another work by Daubert. Even the large rampart is probably an enclosure laid out by the same forester.
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Severely damaged by World War II bombing, the church was subsequently restored and became a Grade I listed building in 1950. Lydd church with its tall tower was a major link in the chain of trigonometric measuring points for the Anglo-French Survey (1784–1790) linking the Royal Greenwich Observatory and the Paris Observatory. This eighteenth-century survey was led by General William Roy, and included a secondary base-line for checking purposes on Romney Marsh, between Ruckinge and Dymchurch. The primary base-line was on Hounslow Heath.
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This relation extended to stars which lay away from the ecliptic and could not be occulted by the Moon, as well as to Cepheid variables, yielding their distances. The relation between angular diameter and V-R colour index is termed the Barnes- Evans Relation, which is calibrated by using direct diameter observations of Cepheid variables. Using these relations the distance is calculated to delta Cephei, and compared with an independent distance derived from trigonometric parallax measurements by the Hubble Space Telescope - the two measurements agree to within a few percent.
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Before the northern aspect of the Ben Lomond plateau was surveyed, Stacks Bluff (at the plateau's southern extremity) was thought to be the highest elevation on the Ben Lomond plateau. From 1905 to 1912 a full survey of Ben Lomond was conducted by Colonel W.V. Legge and his survey party. The survey party explored the highlands on the north of the plateau in 1907. Legge had long suspected that the north of the plateau was higher than the trigonometric station on Stacks Bluff although it is less obviously elevated from casual observation.
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Screw threads are almost never made perfectly sharp (no truncation at the crest or root), but instead are truncated, yielding a final thread depth that can be expressed as a fraction of the pitch value. The UTS and ISO standards codify the amount of truncation, including tolerance ranges. A perfectly sharp 60° V-thread will have a depth of thread ("height" from root to crest) equal to 0.866 of the pitch. This fact is intrinsic to the geometry of an equilateral triangle — a direct result of the basic trigonometric functions.
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1858 triangulation of Mount Everest In 1856 the Great Trigonometric Survey was able to calculate that the highest peak in the world was not Kangchenjunga, but the somewhat unnoticed Peak XV, measured to be 29,002 feet high.The nearest observation was from a distance of . Modern techniques would have measured the height at that time to be about 29,030 feet. Mountaineering was in its infancy but antagonism and apathy towards it were waning so, by 1907, to celebrate the fiftieth anniversary of the Alpine Club, a definite plan was hatched for a British reconnaissance of Everest.
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In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results-series expansion for trigonometric functions-were described in a Sanskrit verse in a book by Somayaji called Tantrasangraha, and again in a commentary called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.
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However, there is a Young Road in Dehradun on which ONGC's Tel Bhawan stands. In 1832, Mussoorie was the intended terminus of the Great Trigonometric Survey of India that began at the southern tip of the country. Although unsuccessful, the Surveyor General of India at the time, George Everest, wanted the new office of the Survey of India to be based in Mussoorie; a compromise location was Dehradun, where it remains. The same year the first beer brewery at Mussoorie was established by Sir Henry Bohle as "The Old Brewery".
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This was an immense undertaking which involved the solution of 920 equations without the aid of matrix methods or digital computers.The calculations were simplified by splitting the mesh into 21 small sub-meshes. The only available computers were the living personnel of the Trigonometric Section, twenty one of them. Once the triangles had been fixed it was then possible to calculate all the sides of the mesh in terms of the length of either of the bases, one by Lough Foyle in Ireland and the other on Salisbury plain.
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In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable . These identities are known collectively as the tangent half-angle formulae because of the definition of . These identities can be useful in calculus for converting rational functions in sine and cosine to functions of in order to find their antiderivatives. Technically, the existence of the tangent half-angle formulae stems from the fact that the circle is an algebraic curve of genus 0.
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This is in part also due to the complex nature of some trigonometric calculations which would be comparably difficult to perform on a conventional scientific calculator. The graphic nature of the flight computer also helps catching many errors which in part explains their continued popularity. The ease of use of electronic calculators means typical flight training literature does not cover the use of calculators or computers at all. In the ground exams for numerous pilot ratings, programmable calculators or calculators containing flight planning software are permitted to be used.
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When the Pulitzer Arts Foundation decided to publish a book about the series, Sugimoto asked Jonathan Safran Foer, whom he had met years earlier, to write a text to accompany the nineteen selected photographs. A 2004 series comprises large photographs of antique mathematical and mechanical models, which Sugimoto came across in Tokyo and shot from slightly below.Michael Kimmelman (27 May 2005), ART IN REVIEW; Hiroshi Sugimoto New York Times. The Mathematical forms – stereometric models in plaster – were created in the 19th century to provide students with a visual understanding of complex trigonometric functions.
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Gal's accurate tables is a method devised by Shmuel Gal to provide accurate values of special functions using a lookup table and interpolation. It is a fast and efficient method for generating values of functions like the exponential or the trigonometric functions to within last-bit accuracy for almost all argument values without using extended precision arithmetic. The main idea in Gal's accurate tables is a different tabulation for the special function being computed. Commonly, the range is divided into several subranges, each with precomputed values and correction formulae.
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Polygonometry was a significant part of Lexell's work. He used the trigonometric approach using the advance in trigonometry made mainly by Euler and presented a general method of solving simple polygons in two articles "On solving rectilinear polygons". Lexell discussed two separate groups of problems: the first had the polygon defined by its sides and angles, the second with its diagonals and angles between diagonals and sides. For the problems of the first group Lexell derived two general formulas giving n equations allowing to solve a polygon with n sides.
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He then withdrew to Thury, where he died in 1845. In 1770, he published an account of a voyage to America in 1768, undertaken as the commissary of the French Academy of Sciences with a view to testing Pierre Le Roy’s watches at sea. In 1783, he sent a memoir to the Royal Society in which he proposed a trigonometric survey connecting the observatories of Paris and Greenwich for the purpose of better determining the latitude and longitude of the latter. His proposal was accepted, resulting in the Anglo-French Survey (1784–1790).
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The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom. An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions.
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Modern mathematical analysis is the study of infinite processes. It is the branch of mathematics that includes calculus. It can be applied in the study of classical concepts of mathematics, such as real numbers, complex variables, trigonometric functions, and algorithms, or of non-classical concepts like constructivism, harmonics, infinity, and vectors. Florian Cajori explains in A History of Mathematics (1893) the difference between modern and ancient mathematical analysis, as distinct from logical analysis, as follows: > The terms synthesis and analysis are used in mathematics in a more special > sense than in logic.
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The TI-32 Math Explorer Plus is a calculator by Texas Instruments specifically designed for middle school students. The Math Explorer Plus was offered as a more advanced version of the TI-12 Math Explorer. The TI-32 Math Explorer Plus offered trigonometric, exponential, logarithmic, and probability functions, and thus can be considered a true scientific calculator unlike the TI-12 Math Explorer. The Math Explorer Plus was eventually replaced by the TI-34 II Explorer Plus, which combined features of the TI-32 and TI-34, as well as incorporating a two-line display.
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The toponym 'Stacks Bluff' first appears on maps in 1915. The 'uppermost peak' of the Bluff (the first prominent isolated eminence) was hitherto known locally as Ernest Crag (or Craig), although this name no longer appears on modern maps. In 1841 the plateau was surveyed by the Polish Explorer Strzelecki who incorrectly calculated barometrically the summit of the plateau as being Stacks Bluff at . After a further survey by James Sprent, the peak had a trigonometric survey point and an elaborate summit cairn constructed by convict workers in 1852.
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The East India Company, and later the British Empire, sought to form trade relations with Tibet. Additionally, exploration of Central Asia and Tibet were of particular interest during the Great Trigonometric Survey of India because their geography was largely unknown to the British. However, Qing China closed Tibet's borders after gaining significant control over Tibet's internal politics following the 1791 Sino-Nepalese War. A number of Europeans tried to reach Lhasa from India over the next hundred years; however, few successfully reached it, and most attempts at entry were turned back by local officials.
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The implementations above always draw only complete octants or circles. To draw only a certain arc from an angle \alpha to an angle \beta, the algorithm needs first to calculate the x and y coordinates of these end points, where it is necessary to resort to trigonometric or square root computations (see Methods of computing square roots). Then the Bresenham algorithm is run over the complete octant or circle and sets the pixels only if they fall into the wanted interval. After finishing this arc, the algorithm can be ended prematurely.
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Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometric functions. In a right triangle, the altitude from each acute angle coincides with a leg and intersects the opposite side at (has its foot at) the right-angled vertex, which is the orthocenter.
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Similarly, the trigonometric identity :\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B) can be interpreted as a shorthand for two equations: one with on both sides of the equation, and one with on both sides. The two copies of the sign in this identity must both be replaced in the same way: it is not valid to replace one of them with and the other of them with . In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.
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Uffe Haagerup was born in Kolding, but grew up on the island of Funen, in the small town of Fåborg. The field of mathematics had his interest from early on, encouraged and inspired by his older brother. In fourth grade Uffe was doing trigonometric and logarithmic calculations. He graduated as a student from Svendborg Gymnasium in 1968, whereupon he relocated to Copenhagen and immediately began his studies of mathematics and physics at the University of Copenhagen, again inspired by his older brother who also studied the same subjects at the same university.
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Barren Mountain is a mountain standing approximately , situated as one of the highest points on the Dorrigo Plateau, that is part of the Great Dividing Range, located in the Northern Tablelands and New England regions of New South Wales, Australia. The mountain is located within the New England National Park and is situated east northeast of Majors Point Trigonometric Station and about north of the junction of Andersons and Cooks Creeks. The nearest settlement, , is located away to the east. The nearest sealed road is the Waterfall Way, located to the north.
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Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function for which Fourier inversion holds good can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically.) by :f(t) = \int_0^\infty \bigl( a(\lambda ) \cos( 2\pi \lambda t) + b(\lambda ) \sin( 2\pi \lambda t)\bigr) \, d\lambda. This is called an expansion as a trigonometric integral, or a Fourier integral expansion.
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Aleksei Georgievich Postnikov (Алексей Георгиевич Постников, 12 June 1921, Moscow – 22 March 1995) was a Russian mathematician, who worked on analytic number theory. He is known for the Postnikov character formula, which expresses the value of a Dirichlet character by means of a trigonometric function of a polynomial with rational coefficients. Postnikov's father was a high-ranking economic functionary who was arrested in 1938 and became a victim of Stalin's purges. Alexei Postnikov studied from 1939 at the Lomonosov University, interrupted by WW II, so that his degree was delayed until 1946.
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Bressoud, David. 2002. "Was Calculus Invented in India?" The College Mathematics Journal (Mathematical Association of America). 33(1):2–13. The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: where, for r = 1, the series reduce to the standard power series for these trigonometric functions, for example: (The Kerala school did not use the "factorial" symbolism.) The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results.
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More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group G(AF), for an algebraic group G and an algebraic number field F, is a complex- valued function on G(AF) that is left invariant under G(F) and satisfies certain smoothness and growth conditions. Poincaré first discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through the Langlands conjectures automorphic forms play an important role in modern number theory.
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The method has only ever been used for a small number of clusters. This is because for the method to work, the cluster must be quite close to Earth (within a few hundred parsecs), and also be fairly tightly bound so it can be made out on the sky. Also, the method is quite difficult to work with compared with more straightforward methods like trigonometric parallax. Finally, the uncertainty in the final distance values are in general fairly large compared those obtained with precision measurements like those from Hipparcos.
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MathomaticFSF Free Software Directory entry is a free, portable, general- purpose computer algebra system (CAS) that can symbolically solve, simplify, combine, and compare algebraic equations, and can perform complex number, modular, and polynomial arithmetic, along with standard arithmetic. It does some symbolic calculus (derivative, extrema, Taylor series, and polynomial integration and Laplace transforms), numerical integration, and handles all elementary algebra except logarithms. Trigonometric functions can be entered and manipulated using complex exponentials, with the GNU m4 preprocessor. Not currently implemented are general functions like f(x), arbitrary-precision and interval arithmetic, and matrices.
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Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if is far from . That is, the Taylor series diverges at if the distance between and is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.
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Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: , , , where is the common measure of five acute angles. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.
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A holonomic function, also called a D-finite function, is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients. Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include polynomials, algebraic functions, logarithm, exponential function, sine, cosine, hyperbolic sine, hyperbolic cosine, inverse trigonometric and inverse hyperbolic functions, and many special functions such as Bessel functions and hypergeometric functions. Holonomic functions have several closure properties; in particular, sums, products, derivative and integrals of holonomic functions are holonomic.
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The Botanical Garden of the University of Latvia has several departments, which are further divided into sectors, laboratories. It deals with the development of the knowledge base of ornamental horticulture and landscaping, plant introduction and experimental research. In the territory of the Botanical Garden there is an Earth observation station of the Institute of Astronomy of the University of Latvia and the geodetic starting point of the Latvian trigonometric network. The collection of plants in the botanical garden consists of about 8,300 taxa, of which about 2,000 are tropical and subtropical plants.
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When Wang Laboratories found that the hp 9100A used an approach similar to the factor combining method in their earlier LOCI-1 (September 1964) and LOCI-2 (January 1965) Logarithmic Computing Instrument desktop calculators, they unsuccessfully accused Hewlett-Packard of infringement of one of An Wang's patents in 1968. John Stephen Walther at Hewlett-Packard generalized the algorithm into the Unified CORDIC algorithm in 1971, allowing it to calculate hyperbolic functions, natural exponentials, natural logarithms, multiplications, divisions, and square roots. The CORDIC subroutines for trigonometric and hyperbolic functions could share most of their code.
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Fig. 1 Isosceles skinny triangle A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians. The solution is particularly simple for skinny triangles that are also isosceles or right triangles: in these cases the need for trigonometric functions or tables can be entirely dispensed with. The skinny triangle finds uses in surveying, astronomy, and shooting.
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The skinny triangle is useful in gunnery in that it allows a relationship to be calculated between the range and size of the target without the shooter needing to compute or look up any trigonometric functions. Military and hunting telescopic sights often have a reticle calibrated in milliradians, in this context usually called just mils or mil-dots. A target in height and measuring in the sight corresponds to a range of 1000 metres. There is an inverse relationship between the angle measured in a sniper's sight and the distance to target.
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About the same time he revived the Sumner method of finding a ship's position, and calculated a set of tables for its ready application. In 1876, he constructed a harmonic analyzer, in which an assembly of disks were used to sum trigonometric series and thus to predict tides. Kelvin mentioned that a similar device could be built to solve differential equations. During the 1880s, Thomson worked to perfect the adjustable compass to correct errors arising from magnetic deviation owing to the increased use of iron in naval architecture.
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Although Mount Krim is a prominent feature against the Ljubljana Marsh, it did not play a significant role in the past. In 1817, geodesists set up a first-class trigonometric point, numbered 172, on the mountain, which was used as a benchmark coordinate for cadastral measures for Carniola, the Littoral, and Istria. Measurements were taken from 1817 to 1828, and this is commemorated by a bronze memorial plaque on the wall of the mountain lodge."Spominsko obeležje geodetski točki na Krimu." 1994. Naša komuna 32(18/19): 3.
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1986 edition of TI-35 PLUS TI-36 SOLAR, first edition It can display 10 digits mantissa with 2 digits exponent, and calculates with 12-digit precision internally. TI-36 SOLAR was based on 1985 version of TI-35 PLUS, but incorporates solar cells. It addition to standard features such as trigonometric functions, exponents, logarithm, and intelligent order of operations found in TI-30 and TI-34 series of calculators, it also include base (decimal, hexadecimal, octal, binary) calculations, complex values, statistics. Conversions include polar- rectangular coordinates (P←→R), angles.
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A trigonometric improvement by Willebrord Snell (1621) obtains better bounds from a pair of bounds gotten from the polygon method. Thus, more accurate results were obtained from polygons with fewer sides. Viète's formula, published by François Viète in 1593, was derived by Viète using a closely related polygonal method, but with areas rather than perimeters of polygons whose numbers of sides are powers of two. The last major attempt to compute by this method was carried out by Grienberger in 1630 who calculated 39 decimal places of using Snell's refinement.
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This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: : y=f(a)+f'(a)(x-a).\, Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations.
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Scientific mode supports exponents and trigonometric functions, and programmer mode gives the user access to more options related to computer programming. The Calculator program has a long history going back to the very beginning of the Macintosh platform, where a simple four-function calculator program was a standard desk accessory from the earliest system versions. Though no higher math capability was included, third-party developers provided upgrades, and Apple released the Graphing Calculator application with the first PowerPC release (7.1.2) of the Mac OS, and it was a standard component through Mac OS 9.
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During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).
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In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a form suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi. He also gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg also gives accurate tables of sines and tangents correct to 8 decimal places around the same time.
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In 1342, Levi ben Gershon, known as Gersonides, wrote On Sines, Chords and Arcs, in particular proving the sine law for plane triangles and giving five-figure sine tables. A simplified trigonometric table, the "toleta de marteloio", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venetian captain Andrea Bianco. Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,Boyer, p.
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This way a trigonometric equation system is solved, and the muzzle's 3D position relative to the screen is calculated. Then, by projecting the muzzle on the screen with the measured angles the impact point is determined. An early example of this technology (though not using IR) can be seen in the NES Power Glove Accessory, which used three ultrasonic sensors serving the same function as the IR emitters used in some lightguns. A simpler variant is commonly used in arcades, where there are no angle detectors but 4 IR sensors.
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In the 1980s, because of its good location, the tower was used by radio hams of the then Halle district contest team, including the amateur ham callsign "Y34H", used by numerous international amateur radio competitions. This tower was not built as a viewing tower, but as the site for a raised trigonometric sign above a trig point (TP) of the first order by the state survey and, for that reason, used as an observation tower. As a result of advances in technology this trig sign - like others in the other federal states - lost its significance.
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His major work was Thesaurus Logarithmorum Completus (Treasury of all Logarithms) that was first published 1794 in Leipzig (its 90th edition was published in 1924). This mathematical table was actually based on Adriaan Vlacq's tables, but corrected a number of errors and extended the logarithms of trigonometric functions for the small angles. An engineer, Franc Allmer, honourable senator of the Graz University of Technology, has found Vega's logarithmic tables with 10 decimal places in the Museum of Carl Friedrich Gauss in Göttingen. Gauss used this work frequently and he has written in it several calculations.
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The light-year unit appeared a few years after the first successful measurement of the distance to a star other than the Sun, by Friedrich Bessel in 1838. The star was 61 Cygni, and he used a heliometer designed by Joseph von Fraunhofer. The largest unit for expressing distances across space at that time was the astronomical unit, equal to the radius of the Earth's orbit or In those terms, trigonometric calculations based on 61 Cygni's parallax of 0.314 arcseconds, showed the distance to the star to be astronomical units ( or ). Bessel added that light takes 10.3 years to traverse this distance.
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Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note.
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By the end of that year he had completed measurements at all but two of the trigonometric stations. Many of the measurements, particularly the cross channel sightings, were taken at night using intense flares (handled by the artillery). Others required the placing of the instrument on church towers, or even on scaffolded steeples, and in their absence it was sometimes necessary to use a specially constructed portable tower some 30 feet high. The final report of 1790 presents figures for the distance between Paris and Greenwich as well as the precise latitude, longitude and height of the British triangulation stations.
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In the U.S., the SAT Subject Test in Mathematics Level 2 (formerly known as Math II or Math IIC, the "C" representing the sanctioned use of a calculator) is a one-hour multiple choice test. The questions cover a broad range of topics. Approximately 10-14% of questions focus on numbers and operations, 48-52% focus on algebra and functions, 28-32% focus on geometry (coordinate, three-dimensional, and trigonometric geometry are covered; plane geometry is not directly tested), and 8-12% focus on data analysis, statistics and probability. Compared to Mathematics 1, Mathematics 2 is more advanced.
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Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al- Būzjānī or Abū al-Wafā Būzhjānī () (10 June 940 – 15 July 998) was a Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetics for businessmen contains the first instance of using negative numbers in a medieval Islamic text. He is also credited with compiling the tables of sines and tangents at 15 ' intervals. He also introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc.
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The quadrature oscillator uses two cascaded op-amp integrators in a feedback loop, one with the signal applied to the inverting input or two integrators and an invertor. The advantage of this circuit is that the sinusoidal outputs of the two op-amps are 90° out of phase (in quadrature). This is useful in some communication circuits. It is possible to stabilize a quadrature oscillator by squaring the sine and cosine outputs, adding them together, (Pythagorean trigonometric identity) subtracting a constant, and applying the difference to a multiplier that adjusts the loop gain around an inverter.
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Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which for a central angle in a circle gives the length of the straight line segment between the points where the angle intersects the circle. He computed this for a circle with a circumference of 21,600 units and a radius (rounded) of 3438 units; this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°.
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In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the Green's function, in the Laplace transform, and in Riemann's theory of trigonometric series, which were not necessarily the Fourier series of an integrable function. These were disconnected aspects of mathematical analysis at the time. The intensive use of the Laplace transform in engineering led to the heuristic use of symbolic methods, called operational calculus. Since justifications were given that used divergent series, these methods had a bad reputation from the point of view of pure mathematics.
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Both parents' families were middle-class, but they sank into poverty after the October revolution of 1917. Ilya became interested in mathematics at the age of 10, struck, as he wrote in his short memoir, "by the charm and unusual beauty of negative numbers", which his father, a PhD in chemical engineering, showed him. In 1952, Piatetski-Shapiro won the Moscow Mathematical Society Prize for a Young Mathematician for work done while still an undergraduate at Moscow University. His winning paper contained a solution to the problem of the French analyst Raphaël Salem on sets of uniqueness of trigonometric series.
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The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for \sin\left(a + b\right) and \sin\left(a - b\right) .
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All trigonometric polynomials are holomorphic on a whole complex plane, and there is a simple theorem in complex analysis that says that all zeros of non-constant holomorphic function are isolated. But this contradicts our earlier finding that F_2 has intervals full of zeros, because points in such intervals are not isolated. Thus the only time- and bandwidth-limited signal is a constant zero. One important consequence of this result is that it is impossible to generate a truly bandlimited signal in any real-world situation, because a bandlimited signal would require infinite time to transmit.
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Map projection formulas depend on the geometry of the projection as well as parameters dependent on the particular location at which the map is projected. The set of parameters can vary based on the type of project and the conventions chosen for the projection. For the transverse Mercator projection used in UTM, the parameters associated are the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor. Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator are a complex mix of algebraic and trigonometric functions.
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He turned next to trigonometric functions with his 1903 paper "Sur les séries trigonométriques". He presented three major theorems in this work: that a trigonometrical series representing a bounded function is a Fourier series, that the nth Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series is integrable term by term. In 1904-1905 Lebesgue lectured once again at the Collège de France, this time on trigonometrical series and he went on to publish his lectures in another of the "Borel tracts". In this tract he once again treats the subject in its historical context.
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In 1542, Rheticus on the recommendation of Joachim Camerarius in conjunction with Melanchthon was then appointed professor of higher mathematics at Leipzig. Rheticus ended up taking another leave of absence in 1545, departing for Italy although the specifics of his itinerary remain unknown. In 1546-7, he would suffer from some unspecified severe mental disorder in Lindau, but recovered enough to return to teaching at Constance towards the latter. By 1551, he would publish some of his work in mathematics, trigonometric tables containing all six functions defined directly in terms of right triangles instead of circles, the first of its kind.
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The interval [−1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine and the hyperbolic function tanh. This interval may be used for the domain of inverse functions. For instance, when θ is restricted to [−π/2, π/2] then sin(θ) is in this interval and arcsine is defined there. Sometimes, the term "unit interval" is used to refer to objects that play a role in various branches of mathematics analogous to the role that [0,1] plays in homotopy theory.
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The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system. ;Clock position (n = 12): A clock position is the relative direction of an object described using the analogy of a 12-hour clock.
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Two types of punched cards used to program the machine. Foreground: 'operational cards', for inputting instructions; background: 'variable cards', for inputting data Babbage's first attempt at a mechanical computing device, the Difference Engine, was a special-purpose machine designed to tabulate logarithms and trigonometric functions by evaluating finite differences to create approximating polynomials. Construction of this machine was never completed; Babbage had conflicts with his chief engineer, Joseph Clement, and ultimately the British government withdrew its funding for the project. During this project, Babbage realised that a much more general design, the Analytical Engine, was possible.
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The nomenclature of features on the mountain were established at this time and were named after members of the survey party and famous explorers of the period. The survey party explored the highlands on the north of the plateau in 1907. Legge had long suspected that the north of the plateau was higher than the trigonometric station on Stacks Bluff (called by him Ben Lomond Fell or Bluff) but was less obviously elevated. Moreover, the area was, at the time, an area so remote and unexplored that Legge described it as 'untrodden as the distant ranges of the west coast'.
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One must loop over each pair of elements (so we get n2 interactions) and for each pair of elements we loop through Gauss points in the elements producing a multiplicative factor proportional to the number of Gauss-points squared. Also, the function evaluations required are typically quite expensive, involving trigonometric/hyperbolic function calls. Nonetheless, the principal source of the computational cost is this double-loop over elements producing a fully populated matrix. The Green's functions, or fundamental solutions, are often problematic to integrate as they are based on a solution of the system equations subject to a singularity load (e.g.
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They manipulated the error term to derive a faster converging series for \pi. They used the improved series to derive a rational expression, 104348/33215 for \pi correct up to nine decimal places, i.e. 3.141592653 (of 3.1415926535897...).Roy, 291–306 The development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) was carried out by mathematicians of the Kerala School in the 15th century CE. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).
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Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics.
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Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Gottfried Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.Struik (1969), 367 In the 18th century, "function" lost these geometrical associations. Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination.
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Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.. Ptolemy's Almagest, a treatise on mathematical astronomy written in the second century AD, uses base 60 to express the fractional parts of numbers. In particular, his table of chords, which was essentially the only extensive trigonometric table for more than a millennium, has fractional parts of a degree in base 60\. Medieval astronomers also used sexagesimal numbers to note time. Al-Biruni first subdivided the hour sexagesimally into minutes, seconds, thirds and fourths in 1000 while discussing Jewish months.
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Bari's decision to travel may have been influenced by the disintegration of the Luzitanians. Luzin's irascible, demanding personality had alienated many of the mathematicians who had gathered around him. By 1930, all traces of the Luzitania movement had vanished, and Luzin left Moscow State for the Academy of Science's Steklov Institute of Mathematics. In 1932, she became a professor at Moscow State University and in 1935 was awarded the title of Doctor of Physical and Mathematical Sciences, a more prestigious research degree than traditional Ph.D. By this time, she had completed foundational work on trigonometric series.
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In October 1871, St John was sent to Baluchistan for the survey of the Perso-Kelat frontier. He returned to England in October 1872 and worked on preparing maps at the India Office. These maps were based on longitudes of the Persian telegraph stations fixed in co-operation with General James Walker of the Indian Trigonometric Survey, Captain William Pierson, RE, and Lt Stiffe, IN. St John published his notes in the Narrative of a Journey through Baluchistan and Southern Persia (1876). He returned to India in 1875 and became principal of the Mayo College in Ajmer.
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In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala School of Astronomy and Mathematics further expanded his works with various series expansions and rational approximations until the 16th century. In the 17th century, James Gregory also worked in this area and published several Maclaurin series.
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One example was the Microsoft BASIC interpreter supplied with most microcomputers that ran VisiCalc. This allowed skilled BASIC programmers to add features, such as trigonometric functions, that VisiCalc lacked. Bricklin and Frankston originally intended to fit the program into 16k memory, but they later realized that the program needed at least 32k. Even 32k was too small to support some features that the creators wanted to include, such as a split text/graphics screen. However, Apple eventually began shipping all Apple IIs with 48k memory following a drop in RAM prices, which enabled the developers to include more features.
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Such a survey had been proposed by William Roy (1726–1790) on his completion of the Anglo-French Survey but it was only after his death that the Board of Ordnance initiated the trigonometric survey, motivated by military considerations in a time of a threatened French invasion. Most of the work was carried out under the direction of Isaac Dalby, William Mudge and Thomas Frederick Colby, but the final synthesis and report (1858) was the work of Alexander Ross Clarke. The survey stood the test of time for a century, until the Retriangulation of Great Britain between 1935 and 1962.
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The two polynomials, and with , are orthogonal, if and only if, In other words, for arbitrary parameters, only a finite number of Romanovski polynomials are orthogonal. This property is referred to as finite orthogonality. However, for some special cases in which the parameters depend in a particular way on the polynomial degree infinite orthogonality can be achieved. This is the case of a version of equation () that has been independently encountered anew within the context of the exact solubility of the quantum mechanical problem of the trigonometric Rosen–Morse potential and reported in Compean & Kirchbach (2006).
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In the 1730s, he first established and used what was later to be known as Catalan numbers.The 18th century Chinese discovery of the Catalan numbers The Jesuit missionaries' influence can be seen by many traces of European mathematics in his works, including the use of Euclidean notions of continuous proportions, series addition, subtraction, multiplication and division, series reversion, and the binomial theorem. Minggatu's work is remarkable in that expansions in series, trigonometric and logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus. In 1742 he participated in the revision of the Compendium of Observational and Computational Astronomy.
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Trotter attended Addiscombe Military Seminary from 1858 to 1860, and was awarded his commission in the Royal Engineers, Bengal on 8 June 1860. He sailed to India in 1862, and from 1863 to 1875 served on the Great Trigonometric Survey. He was a member of the Second Yarkand Mission to Sinkiang to visit the territory ruled by Yakub Beg: the mission had 350 support staff and 6,476 porters, and was led by Sir Thomas Douglas Forsyth. Among the other Indian Army officers were Thomas E. Gordon, John Biddulph, Henry Bellew, Ferdinand Stoliczka and R. A. Champman.
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The sliding bead facilitates trigonometric calculations with the instrument. Traditionally the line from the beginning of the arc to the apex is called “Jaibs” and the line from the end of the arc to the apex is called “Jaib tamams”. Both jaibs and jaib tamams are divided into 60 equal units and the sixty parallel lines to the jaibs are called sitheeniys or” sixtys “ and the sixty parallel lines to the jaib tamams are “juyoobul mabsootah”. The reason for sixty divisions along the Jaibs and Jaib Tamams is that the instrument uses the Sexagesimal number system.
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A ray through the unit hyperbola x^2\ -\ y^2\ =\ 1 at the point (\cosh\,a,\,\sinh\,a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative. Just as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram. In a unit circle, the angle (in radians) is equal to twice the area of the circular sector which that angle subtends.
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Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (1820–23) also attacked the problem from a different standpoint. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and du Bois-Reymond.
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Thus C=2\pi R=\pi D is seen to be true as a theorem. Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula A=\pi r^2, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can develop trigonometric functions and the fundamental constant in a way that is totally independent of their relation to geometry. We have indicated where appropriate how each of these proofs can be made totally independent of all trigonometry, but in some cases that requires more sophisticated mathematical ideas than those afforded by elementary calculus.
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A unique feature of the Mark II is that it had built- in hardware for several functions such as the reciprocal, square root, logarithm, exponential, and some trigonometric functions. These took between five and twelve seconds to execute. The Mark I and Mark II were not a stored- program computer – it read an instruction of the program one at a time from a tape and executed it (like the Mark I). This separation of data and instructions is known as the Harvard architecture. The Mark II had a peculiar programming method that was devised to ensure that the contents of a register were available when needed.
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Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhāṣā (c.1500–1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).
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Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter to denote the imaginary unit. The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones.
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The tondo e quadro (circle and square) from Andrea Bianco's 1436 atlas The rule of marteloio is a medieval technique of navigational computation that uses compass direction, distance and a simple trigonometric table known as the toleta de marteloio. The rule told mariners how to plot the traverse between two different navigation courses by means of resolving triangles with the help of the Toleta and basic arithmetic. Those uncomfortable with manipulating numbers could resort to the visual tondo e quadro (circle-and-square) and achieve their answer with dividers. The rule of marteloio was commonly used by Mediterranean navigators during the 14th and 15th centuries, before the development of astronomical navigation.
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At various times, her governesses were native speakers of English, French, and German. When she was 8 or 9 years old, she was intrigued by a foretaste of what she was to learn later in her lessons in calculus; the wall of her room had been papered with pages from lecture notes by Ostrogradsky, left over from her father's student days. She was tutored privately in elementary mathematics by Iosif Ignatevich Malevich. The physicist Nikolai Nikanorovich Tyrtov noted her unusual aptitude when she managed to understand his textbook by discovering for herself an approximate construction of trigonometric functions which she had not yet encountered in her studies.
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Nasīr al-Dīn al- Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.
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The village continues northwest towards the shall hills of Rovovi, Mala Hrastovača and thus the borders finish at the Lipova Glava. From there the borders turn north to the hill of Bogdanica and through the most forest area of Krestelovac they emerge at the Šabanova Glavica hill. The borders continue Northwest at the Vuchicke Bare and over the hill of Puskarnica reaches the Grualj hill. From the Grualj the borderline is directed from the Kraljicino Brdo to the Stupa read, and continue to the vines (Veliki Vinogradi) and Prokop, then from the small hills of Velika and Mala Balabanovića and reaches the trigonometric spot 249.
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Published by the Survey of Nepal, this is Map 50 of the 57 map set at 1:50,000 scale "attached to the main text on the First Joint Inspection Survey, 1979–80, Nepal-China border." At the top centre, a boundary line, identified as separating "China" and "Nepal", passes through the summit contour. The boundary here and for much of the China–Nepal border follows the main Himalayan watershed divide. Kangshung Face (the east face) as seen from orbit In 1856, Andrew Waugh announced Everest (then known as Peak XV) as high, after several years of calculations based on observations made by the Great Trigonometric Survey.
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At one time, it was common to use spider silk as a thread.Berenbaum, May R., Field Notes — Spin Control, The Sciences, The New York Academy Of Sciences, September/October 1995 By placing one wire over one point of interest and moving the other to a second point, the distance between the two wires can be measured with the micrometer portion of the instrument. Given this precise distance measurement at the image plane, a trigonometric calculation with the objective focal length yields the angular distance between the two points seen in a telescope. In a microscope, a similar calculation yields the spatial distance between two points on a specimen.
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24 Scorpii is a star that was originally placed by John Flamsteed within the constellation of Scorpius but in now placed within the southeastern constellation of Ophiuchus. It is visible to the naked eye as a faint, yellow- hued point of light with an apparent visual magnitude of 4.91. Based on the trigonometric parallax published in Gaia Data Release 2, the star lies approximately 121 parsecs or 390 light years away. It is positioned near the ecliptic and thus is subject to lunar occultations. This object is a luminous giant star that is classified by spectral and luminosity class as G7.5II or G7.5II-IIICN1Ba0.5.
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Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions and are orthogonal on the interval x \in (-\pi, \pi) when m eq n and n and m are positive integers. For then :2 \sin (mx) \sin (nx) = \cos \left((m - n)x\right) - \cos\left((m+n) x\right), and the integral of the product of the two sine functions vanishes.Antoni Zygmund (1935) Trigonometrical Series, page 6, Mathematical Seminar, University of Warsaw Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.
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Iriññāttappiḷḷi Mādhavan Nampūtiri known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Aloor, Irinjalakuda in Thrissur District, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".
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Records of Moore's life during the next ten years are sketchy, but by 1650 he was an established mathematics teacher and published his first book, Moores Arithmetick. In 1674, Sir Jonas Moore first used the abbreviated notation 'cos' for the trigonometric term cosine. He went on that year to be appointed Surveyor to the Fen drainage Company of William Russell, 5th Earl of Bedford, and worked on draining the Fens for the next seven years. In 1658, Moore was able to produce a 16-sheet Mapp of the Great Levell of the Fens, which provided an effective means of displaying the Company's achievements in altering the Fenland landscape of East Anglia.
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The BPM may encounter difficulty in the solution of problems having complex source functions, such as non-smooth, large-gradient functions, or a set of discrete measured data. The solution of such problems involves: (1) The complex functions or a set of discrete measured data can be interpolated by a sum of polynomial or trigonometric function series. Then, the RC-MRM can reduce the nonhomogeneous equation to a high-order homogeneous equation, and the BPM can be implemented to solve these problems with boundary-only discretization. (2) The domain decomposition may be used to in the BPM boundary-only solution of large-gradient source functions problems.
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Sample configurations of pentagramma mirificum Napier’s circles contain circular shifts of parts (a, \pi/2-B, \pi/2-c, \pi/2-A, b) Pentagramma mirificum (Latin for miraculous pentagram) is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici Logarithmorum Canonis Descriptio (Description of the Admirable Table of Logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). The properties of pentagramma mirificum were studied, among others, by Carl Friedrich Gauss.
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In 1875 Smith published the important paper on the integrability of discontinuous functions in Riemann's sense.See . In this work, while giving a rigorous definition of the Riemann integral as well as explicit rigorous proofs of many of the results published by Riemann,The Riemann integral was introduced in Bernhard Riemann's paper "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series), submitted to the University of Göttingen in 1854 as Riemann's Habilitationsschrift (qualification to become an instructor). It was published in 1868 in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Proceedings of the Royal Philosophical Society at Göttingen), vol.
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Baxter R J, Perk J' H' H' and Au-Yang H (1988), "New solutions of the star-triangle relations for the chiral Potts model", Physics Letters A 128 138–42. Unlike the other solvable models,R. J. Baxter,"Exactly Solved Models in Statistical Mechanics", Academic Press, .B. M. McCoy, "Advanced Statistical Mechanics", 146 International Series of Monographs on Physics, Oxford, England, whose weights are parametrized by curves of genus less or equal to one, so that they can be expressed in term of trigonometric, or rational function (genus=0) or by theta functions (genus=1), this model involves high genus theta functions, which are not yet well developed.
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Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
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Steinhaus authored over 170 works. Unlike his student, Stefan Banach, who tended to specialize narrowly in the field of functional analysis, Steinhaus made contributions to a wide range of mathematical sub-disciplines, including geometry, probability theory, functional analysis, theory of trigonometric and Fourier series as well as mathematical logic. He also wrote in the area of applied mathematics and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kac's words, "even lawyers". Probably his most notable contribution to functional analysis was the 1927 proof of the Banach–Steinhaus theorem, given along with Stefan Banach, which is now one of the fundamental tools in this branch of mathematics.
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One man whose name is internationally famous is also buried in a grave on the south side of the church. Sir George Everest, the geographer who undertook the Great Trigonometric Survey in India while acting as Surveyor-General, was the first person to determine the exact height of the world's highest mountain, which was then named after him. He died in London in December 1866 and was buried with his two children, sister and father-in-law. However, Sir George himself had no connection with Hove or Brighton at any time during his life, and none of the family members buried at the church were known to be associated with it.
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In 1813 the Duchy was restored to the Electorate of Hanover, which - after its upgrade to the Kingdom of Hanover in 1814 (and still ruled in personal union with the United Kingdom of Great Britain and Ireland) - incorporated the Duchy in a real union and the ducal territory, including Zeven, became part of the Stade Region, established in 1823. In 1824/1825 Carl Friedrich Gauß carried out the trigonometric geodetic surveying of the Stade Region, as commissioned by King George IV of Hanover and the United Kingdom. Gauß used the tower of the Lutheran Church of St. Viti as a benchmark for his triangulations.
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Abu Rayhan al-Biruni (973–1048) gave an estimate of 6,339.6 km for the Earth radius, which is only 17.15 km less than the modern value of 6,356.7523142 km (WGS84 polar radius "b"). In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, Al-Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.Lenn Evan Goodman (1992), Avicenna, p. 31, Routledge, . (cf.
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Calculating trigonometric functions can substantially slow a computing application. The same application can finish much sooner when it first precalculates the sine of a number of values, for example for each whole number of degrees (The table can be defined as static variables at compile time, reducing repeated run time costs). When the program requires the sine of a value, it can use the lookup table to retrieve the closest sine value from a memory address, and may also interpolate to the sine of the desired value, instead of calculating by mathematical formula. Lookup tables are thus used by mathematics coprocessors in computer systems.
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Map of Nain Singh's exploration of Tibet In May 1877, Singh was awarded the Royal Geographic Society's Patron's Medal "for his great journeys and Surveys in Tibet and along the Upper Brahmaputra, he has determined the position of Lhasa, and positive knowledge of the map of Asia." Henry Yule received the award on Singh's behalf and in his acceptance speech said that "[Singh's] observations have added a larger amount of important knowledge to the map of Asia than those of any other living man." On 27 June 2004, an Indian postage stamp featuring Nain SinghTrigonometrical Survey. midco.net was issued commemorating his role in the Great Trigonometric Survey of India.
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The 3-wire method is also used when high precision is needed to inspect a specific diameter, commonly the pitch diameter, or on specialty threads such as multi- start or when the thread angle is not 60°. Ball-shaped micrometer anvils can be used in similar fashion (same trigonometric relationship, less cumbersome to use). Digital calipers and micrometers can send each measurement (data point) as it occurs to storage or software through an interface (such as USB or RS-232), in which case the table lookup is done in an automated way, and quality assurance and quality control can be achieved using statistical process control.
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Papers three and four are "Fundamental Theorems of Analysis Generalized for Space" and "On the definition of the Trigonometric Functions", which he had presented the previous year in Chicago at the Congress of Mathematicians held in connection with the World's Columbian Exhibition. He follows George Salmon in exhibiting the hyperbolic angle, argument of hyperbolic functions. The fifth paper is "Elliptic and Hyperbolic Analysis" which considers the spherical law of cosines as the fundamental theorem of the sphere, and proceeds to analogues for the ellipsoid of revolution, general ellipsoid, and equilateral hyperboloids of one and two sheets, where he provides the hyperbolic law of cosines. In 1900 Alexander published "Hyperbolic Quaternions"A.
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The sine-only expansion for equally spaced points, corresponding to odd symmetry, was solved by Joseph Louis Lagrange in 1762, for which the solution is a discrete sine transform. The full cosine and sine interpolating polynomial, which gives rise to the DFT, was solved by Carl Friedrich Gauss in unpublished work around 1805, at which point he also derived a fast Fourier transform algorithm to evaluate it rapidly. Clairaut, Lagrange, and Gauss were all concerned with studying the problem of inferring the orbit of planets, asteroids, etc., from a finite set of observation points; since the orbits are periodic, a trigonometric interpolation was a natural choice.
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From 1930 until 1952 he was a member of the Warsaw Scientific Society (TNW), from 1946 of the Polish Academy of Learning (PAU), from 1959 of the Polish Academy of Sciences (PAN), and from 1961 of the National Academy of Sciences in the United States. In 1986 he received the National Medal of Science. In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field".
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In 1970 a British Royal Engineer Survey Troop was attached to 1 Fd Svy Sqn. Vertical (height) control for the mapping was a combination of trigonometric heighting, barometric heighting by helicopter, levelling with connections to sea level and WREMAPS II Airborne Profile Recorder (APR).Sargent, Clem, Lieutenant- Colonel RA Svy, 1990, The Royal Australian Survey Corps 1915–1990Royal Australian Survey Corps Association Bulletin 1965 to 1989School of Military Survey, 1985, The Chronology of RA Svy Corps, Edition 2 There are few countries in the world that present more challenging weather, terrain, vegetation, transport, living, communications, health and volcanic conditions for astronomic and terrestrial geodetic surveys than Papua New Guinea.
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Simon Stevin, in his book describing decimal representation of fractions (De Thiende), cites the trigonometric tables of Regiomontanus as suggestive of positional notation.E. J. Dijksterhuis (1970) Simon Stevin: Science in the Netherlands around 1600, pages 17–19, Martinus Nijhoff Publishers, Dutch original 1943 Regiomontanus designed his own astrological house system, which became one of the most popular systems in Europe. In 1561, Daniel Santbech compiled a collected edition of the works of Regiomontanus, De triangulis planis et sphaericis libri quinque (first published in 1533) and Compositio tabularum sinum recto, as well as Santbech's own Problematum astronomicorum et geometricorum sectiones septem. It was published in Basel by Henrich Petri and Petrus Perna.
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"A History of Elementary Mathematics – With Hints on Methods of Teaching". p.94. More recently, as discussed in the above section, the infinite series of calculus for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described (with proofs and formulas for truncation error) in India, by mathematicians of the Kerala school, remarkably some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China and Arabia, and, from around 1500, with Europe.
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The principal triangulation mesh over Britain. The Principal Triangulation of Britain was the first high-precision trigonometric survey of the whole of Great Britain (including Ireland), carried out between 1791 and 1853 under the auspices of the Board of Ordnance. The aim of the survey was to establish precise geographical coordinates of almost 300 significant landmarks which could be used as the fixed points of local topographic surveys from which maps could be drawn. In addition there was a purely scientific aim in providing precise data for geodetic calculations such as the determination of the length of meridian arcs and the figure of the Earth.
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In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, Abu Rayhan al-Biruni (973–1048) developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded simpler measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.Lenn Evan Goodman (1992), Avicenna, p. 31, Routledge, . (cf. ) Al-Biruni's method's motivation was to avoid "walking across hot, dusty deserts" and the idea came to him when he was on top of a tall mountain in India (present day Pind Dadan Khan, Pakistan).
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The University of Kerala in Thiruvananthapuram The Kerala school of astronomy and mathematics flourished between the 14th and 16th centuries. In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts, including series expansion for trigonometric functions. In the early decades of the 19th century, the modern educational transformation of Kerala was triggered by the efforts of the Church Mission Society missionaries to promote mass education. Following the recommendations of the Wood's despatch of 1854, the princely states of Travancore and Cochin launched mass education drives mainly based on castes and communities, and introduced a system of grant-in-aid to attract more private initiatives.
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James Gregory FRS (November 1638 – October 1675) was a Scottish mathematician and astronomer. His surname is sometimes spelled as Gregorie, the original Scottish spelling. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions. In his book Geometriae Pars Universalis (1668) Gregory gave both the first published statement and proof of the fundamental theorem of the calculus (stated from a geometric point of view, and only for a special class of the curves considered by later versions of the theorem), for which he was acknowledged by Isaac Barrow.
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A twiddle factor, in fast Fourier transform (FFT) algorithms, is any of the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. This term was apparently coined by Gentleman & Sande in 1966, and has since become widespread in thousands of papers of the FFT literature. More specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm, used to recursively combine smaller discrete Fourier transforms. This remains the term's most common meaning, but it may also be used for any data-independent multiplicative constant in an FFT.
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Robson argues on linguistic grounds that the trigonometric theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time. In 2003, the MAA awarded Robson with the Lester R. Ford Award for her work, stating it is "unlikely that the author of Plimpton 322 was either a professional or amateur mathematician. More likely he seems to have been a teacher and Plimpton 322 a set of exercises.". Robson takes an approach that in modern terms would be characterized as algebraic, though she describes it in concrete geometric terms and argues that the Babylonians would also have interpreted this approach geometrically.
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While the notation might be misunderstood, certainly denotes the multiplicative inverse of and has nothing to do with the inverse function of . In keeping with the general notation, some English authors use expressions like to denote the inverse of the sine function applied to (actually a partial inverse; see below) Other authors feel that this may be confused with the notation for the multiplicative inverse of , which can be denoted as . To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus). For instance, the inverse of the sine function is typically called the arcsine function, written as .
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Note that, by the principle of reciprocity, the pattern observed when a particular antenna is transmitting is identical to the pattern measured when the same antenna is used for reception. Typically one finds simple relations describing the antenna far-field patterns, often involving trigonometric functions or at worst Fourier or Hankel transform relationships between the antenna current distributions and the observed far-field patterns. While far-field simplifications are very useful in engineering calculations, this does not mean the near-field functions cannot be calculated, especially using modern computer techniques. An examination of how the near fields form about an antenna structure can give great insight into the operations of such devices.
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It consisted of a brass disk on which a number of circular logarithmic scales were inscribed, with two radial wires that could each be locked to a point on the circumference. Using this instrument, a sailor could perform various trigonometric calculations by setting the wire to the position of the argument on one of the circular scales and reading the result from another of the circular scales. Ayres made fine, large Azimuth compasses, used in determining how much the magnetic compass deviated from true north. A brass mariner's compass in gimbals set in mahogany box, made by Ayres in Amsterdam around 1775, is said to have been the property of Sir Isaac Newton.
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The World Geodetic System WGS84 ellipsoid is now generally used to model the Earth in the UTM coordinate system, which means current UTM northing at a given point can differ up to 200 meters from the old. For different geographic regions, other datum systems can be used. Prior to the development of the Universal Transverse Mercator coordinate system, several European nations demonstrated the utility of grid-based conformal maps by mapping their territory during the interwar period. Calculating the distance between two points on these maps could be performed more easily in the field (using the Pythagorean theorem) than was possible using the trigonometric formulas required under the graticule-based system of latitude and longitude.
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Lieutenant-General James Francis Tennant (10 January 1829 – 6 March 1915) was a noted soldier and astronomer. He was born in Calcutta to Scottish parents. The son of Brigadier-General Sir James Tennant and Elizabeth (née Paterson),Dod, Robert (1862), The Peerage, Baronetage, and Knightage, of Great Britain and Ireland for 1862, London, Whittaker, p.550 he was educated at the East India Company's Military Seminary at Addiscombe from 1845 to 1847, and began his military career with the Bengal Engineers in Calcutta in 1847. His mathematical skills landed him with the Great Trigonometric Survey where he was engaged in triangulation of the great longitudinal series until 1857, when he was diverted to garrison duties during the Indian Mutiny.
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In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence of quadratic equations.. In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have degree over the base field that is a power of two. In the specific case of a regular n-gon, the question reduces to the question of constructing a length :cos , which is a trigonometric number and hence an algebraic number.
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Thus for example, whereas in a regular calculus course students would study trigonometric functions, courses here would not typically cover this area. Correspondingly, these courses typically do not go into the same depth as standard courses in the mathematics or science fields. (Although see Bachelor of Science in Business Administration and Bachelor of Business Science.) Note that economics majors, especially those planning to pursue graduate study in the field, are encouraged to instead take regular calculus, as well as linear algebra and other advanced math courses, especially real analysis. Some economics programs (instead) include a module in "mathematics for economists", providing a bridge between the above "Business Mathematics" courses and mathematical economics and econometrics.
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See, for example, Euclid's algorithm for finding the greatest common divisor of two numbers. By the High Middle Ages, the positional Hindu–Arabic numeral system had reached Europe, which allowed for systematic computation of numbers. During this period, the representation of a calculation on paper actually allowed calculation of mathematical expressions, and the tabulation of mathematical functions such as the square root and the common logarithm (for use in multiplication and division) and the trigonometric functions. By the time of Isaac Newton's research, paper or vellum was an important computing resource, and even in our present time, researchers like Enrico Fermi would cover random scraps of paper with calculation, to satisfy their curiosity about an equation.
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If the catheti of a right triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor (shorter) and major (longer) cathetus. The ratio of the lengths of the catheti defines the trigonometric functions tangent and cotangent of the acute angles in the triangle: the ratio c_1/c_2 is the tangent of the acute angle adjacent to c_2 and is also the cotangent of the acute angle adjacent to c_1. In a right triangle, the length of a cathetus is the geometric mean of the length of the adjacent segment cut by the altitude to the hypotenuse and the length of the whole hypotenuse.
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While trigonometry can be codified—as was clear already to expert mathematicians of the eighteenth century (if not before)—the search for a complete and unified theory of special functions has continued since the nineteenth century. The high point of special function theory in the period 1800–1900 was the theory of elliptic functions; treatises that were essentially complete, such as that of Tannery and Molk, could be written as handbooks to all the basic identities of the theory. They were based on techniques from complex analysis. From that time onwards it would be assumed that analytic function theory, which had already unified the trigonometric and exponential functions, was a fundamental tool.
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In the 1960s, Hewlett-Packard was becoming a diversified electronics company with product lines in electronic test equipment, scientific instrumentation, and medical electronics, and was just beginning its entry into computers. The corporation recognized two opportunities: it might be possible to automate the instrumentation that HP was producing, and HP's customer base were likely to buy a product that could replace the slide rules and adding machines that they were now using for computation. With this in mind, HP built the HP 9100 desktop scientific calculator. This was a full- featured calculator that included not only standard "adding machine" functions but also powerful capabilities to handle floating-point numbers, trigonometric functions, logarithms, exponentiation, and square roots.
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A succession of distance indicators, which is the distance ladder, is needed for determining distances to other galaxies. The reason is that objects bright enough to be recognized and measured at such distances are so rare that few or none are present nearby, so there are too few examples close enough with reliable trigonometric parallax to calibrate the indicator. For example, Cepheid variables, one of the best indicators for nearby spiral galaxies, cannot yet be satisfactorily calibrated by parallax alone, though the Gaia space mission can now weigh in on that specific problem. The situation is further complicated by the fact that different stellar populations generally do not have all types of stars in them.
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Principal lines of spirit levelling in England, 1860 In addition to the determination of the precise location of the trigonometric points the Survey established the precise altitude of a number of fundamental benchmarks by spirit levelling over the years 1839 to 1860. This height survey was completely independent of the position survey, in contrast to modern GPS fixes which give both. The report of the levelling in Ireland had been published in 1855, and it fell to Clarke to prepare the reports for England and Scotland. For England this involved a least squares analysis for the 62 lines of primary levelling with 91 fundamental benchmarks at the end points and at points of intersection.
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The ancient Greek view of the heavenly bodies on which their navigation was based was imported from Babylonia by the Ionian Greeks, who used it to become a seafaring nation of merchants and colonists during the Archaic period in Greece. Massalia was an Ionian colony. The first Ionian philosopher, Thales, was known for his ability to measure the distance of a ship at sea from a cliff by the very method Pytheas used to determine the latitude of Massalia, the trigonometric ratios. The astronomic model on which ancient Greek navigation was based, which is still in place today, was already extant in the time of Pytheas, the concept of the degrees only being missing.
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She thus engendered a lifelong interest in astronomy to young men and women, many of whom were simply satisfying a prerequisite to their undergraduate degrees. During the mid 1950s, Hoffleit consulted for the U.S. Army's Ballistic Research Laboratories in "Doppler reductions." She was the author of the Bright Star Catalogue, a compendium of information on the 9,110 brightest stars in the sky; she also co-authored The General Catalogue of Trigonometric Stellar Parallaxes, containing precise distance measurements to 8,112 stars, information critical to understanding the kinematics of the Milky Way galaxy and the evolution of the solar neighborhood. With Harlan J. Smith, Hoffleit discovered the optical variability of the first-discovered quasar 3C 273.
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The first scientific visit to the Ben Lomond plateau was made by the Polish explorer Paul Edmond de Strzelecki on 28 November 1841 and he measured the height of Stacks Bluff (albeit incorrectly) by barometry as 5002 feet. In 1852; after the site was surveyed by James Sprent, the Government Surveyor, a Trigonometric Point was constructed on Stacks Bluff (the southernmost extremity of Ben Lomond) using convict labour. A Full survey of Ben Lomond was conducted from September 1905 to 1912 by Colonel W.V. Legge and, later, Lyndhurst Giblin took over the survey. The surveyors climbed to the plateau from Mangana but on some of the later visits they ascended via Avoca and the Ben Lomond Marshes.
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A dial plate can be laid out, by a pragmatic approach, observing and marking a shadow at regular intervals throughout the day on each day of the year. If the latitude is known the dial plate can be laid out using geometrical construction techniques which rely on projection geometry, or by calculation using the known formulas and trigonometric tables usually using logarithms, or slide rules or more recently computers or mobile phones. Linear algebra has provided a useful language to describe the transformations. A sundial schema uses a compass and a straight edge to firstly to derive the essential angles for that latitude, then to use this to draw the hourlines on the dial plate.
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The nocturnal revolutionized long distance seafaring by complementing the use of the astrolabe and ephemerides by now giving sailors an accurate tool with which to discover the time at their position. Both Zacuto and Cortes were respected mathematicians and had determined in their respective publications the trigonometric measurements concerning the degrees of latitude and longitude. If a ship's pilot or navigator used the nocturnal to read their time, and then consulted an astrolabe in concert with astronomical charts they could determine the time distance between themselves and a fixed location. The trigonometry, dictated particularly in Zacuto's work, then allowed a sailor to calculate the degree difference east or west of the fixed position.
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PowerPoint includes more templates and transition effects, and OneNote includes a new splash screen. On May 16, 2011, new images of Office 15 were revealed, showing Excel with a tool for filtering data in a timeline, the ability to convert Roman numerals to Arabic numerals, and the integration of advanced trigonometric functions. In Word, the capability of inserting video and audio online as well as the broadcasting of documents on the Web were implemented. Microsoft has promised support for Office Open XML Strict starting with version 15, a format Microsoft has submitted to the ISO for interoperability with other office suites, and to aid adoption in the public sector. This version can read and write ODF 1.2 (Windows only).
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Although Cotes's style was somewhat obscure, his systematic approach to integration and mathematical theory was highly regarded by his peers. Cotes discovered an important theorem on the n-th roots of unity,Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Theoremata tum logometrica tum triogonometrica datarum fluxionum fluentes exhibentia, per methodum mensurarum ulterius extensam" (Theorems, some logorithmic, some trigonometric, which yield the fluents of given fluxions by the method of measures further developed), pages 113-114. foresaw the method of least squares,Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Aestimatio errorum in mixta mathesis per variationes partium trianguli plani et sphaerici" Harmonia mensurarum ... , pages 1-22, see especially page 22.
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Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate.. It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.. It is a special case of the hippopede (lemniscate of Booth), with d=-c, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other. The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate.
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The duration of the planetary revolutions during a mahayuga is given as 4.32 million years. 2\. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetic and geometric progressions; gnomon/shadows (shanku- chhAyA); and simple, quadratic, simultaneous, and indeterminate equations (Kuṭṭaka). 3\. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week. 4\. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc.
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Code in the per_frame section is executed once for each frame, modifying variables which affect different parameters that can be passed to other areas of code. Trigonometric functions which modify MilkDrop's internal looping time variable, systems of logic, and interaction with the audio information received from Winamp or other applicable media player's Fast Fourier transform (FFT) can be used to govern how these parameters evolve through time.Beginners Guide to MilkDrop Preset Writing 28 February 2002. Code in the `per_pixel` section of MilkDrop is not actually re-evaluated at every pixel as the name would suggest, rather the screen is divided into a grid and the code is evaluated at each grid point.
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Accurate calculations of distance based on stellar parallax require a measurement of the distance from Earth to the Sun, now known to exquisite accuracy based on radar reflection off the surfaces of planets.. The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and also the star with the largest parallax), Proxima Centauri, has a parallax of 0.7685 ± 0.0002 arcsec. This angle is approximately that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. A large heliometer was installed at Kuffner Observatory (In Vienna) in 1896, and was used for measuring the distance to other stars by trigonometric parallax.
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An analytic expression (or expression in analytic form) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the th root), logarithms, and trigonometric functions. However, the class of expressions considered to be analytic expressions tends to be wider than that for closed- form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions.
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An animation of the additive synthesis of a square wave with an increasing number of harmonics Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental () have additional nodes. The oscillation seen about the sawtooth when is large is called the Gibbs phenomenon The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string.
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Fig. 2 – Obtuse triangle with perpendicular Though the notion of the cosine was not yet developed in his time, Euclid's Elements, dating back to the 3rd century BC, contains an early geometric theorem almost equivalent to the law of cosines. The cases of obtuse triangles and acute triangles (corresponding to the two cases of negative or positive cosine) are treated separately, in Propositions 12 and 13 of Book 2. Trigonometric functions and algebra (in particular negative numbers) being absent in Euclid's time, the statement has a more geometric flavor: Using notation as in Fig. 2, Euclid's statement can be represented by the formula :AB^2 = CA^2 + CB^2 + 2 (CA)(CH).
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A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space H, in the sense that the closure of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of H, its cardinality is known as the Hilbert space dimension.A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra above.
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The heat equation describes the dissipation of physical properties over time, such as the decline of the temperature of a hot body placed in a colder environment (yellow depicts colder regions than red). Resolving a periodic function into a sum of trigonometric functions forms a Fourier series, a technique much used in physics and engineering.Although the Fourier series is periodic, the technique can be applied to any L2 function on an interval by considering the function to be continued periodically outside the interval. See The underlying vector space is usually the Hilbert space L2(0, 2π), for which the functions sin mx and cos mx (m an integer) form an orthogonal basis.
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Paul Painlevé as a young man Some differential equations can be solved using elementary algebraic operations that involve the trigonometric and exponential functions (sometimes called elementary functions). Many interesting special functions arise as solutions of linear second order ordinary differential equations. Around the turn of the century, Painlevé, É. Picard, and B. Gambier showed that of the class of nonlinear second order ordinary differential equations with polynomial coefficients, those that possess a certain desirable technical property, shared by the linear equations (nowadays commonly referred to as the 'Painlevé property') can always be transformed into one of fifty canonical forms. Of these fifty equations, just six require 'new' transcendental functions for their solution.
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The signal strength, SNR and correlation values are used primarily to determine the quality and accuracy of the velocity data, although the signal strength (acoustic backscatter intensity) may related to the instantaneous suspended sediment concentration with proper calibration. The velocity component is measured along the line connecting the sampling volume to the receiver. The velocity data must be transformed into a Cartesian system of coordinates and the trigonometric transformation may cause some velocity resolution errors. Although acoustic Doppler velocimetry (ADV) has become a popular technique in laboratory in field applications, several researchers pointed out accurately that the ADV signal outputs include the combined effects of turbulent velocity fluctuations, Doppler noise, signal aliasing, turbulent shear and other disturbances.
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Using his improved Hardy-Littlewood method, I. M. Vinogradov published numerous refinements leading to :G(k)\le k(3\log k +11) in 1947 and, ultimately, :G(k)\le k(2\log k +2\log\log k + C\log\log\log k) for an unspecified constant C and sufficiently large k in 1959. Applying his p-adic form of the Hardy-Littlewood-Ramanujan- Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Anatolii Alexeevitch Karatsuba obtained (1985) a new estimate of the Hardy function G(k) (for k \ge 400): : \\! G(k) < 2 k\log k + 2 k\log\log k + 12 k. Further refinements were obtained by Vaughan [1989].
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The trenches and natural barriers are shown in the map reproduced in this section of this Article, which was drawn to scale by Lt. Colonel George Thompson (engineer) of the Paraguayan army; he personally made a detailed trigonometric survey of the ground. The map is corroborated by Burton's detailed verbal description based on his own inspection on horseback and on figures supplied to him by Lt. Colonel Chodasiewicz of the Argentine Army. Burton reported that the layout required a garrison of at least 10,000 men; at the time of the Siege of Humaitá the Allied Commander-in-chief estimated that it had 18,000 and possibly 20,000 men and 120 cannon not including the river batteries.
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The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure". The modern word "sine" is derived from the Latin word sinus, which means "bay", "bosom" or "fold" is indirectly, via Indian, Persian and Arabic transmission, derived from the Greek term khordḗ "bow-string, chord". The Hindu term for sine in Sanskrit is jyā "bow-string", the Hindus originally introduced and usually employed three trigonometric functions jyā, koti-jyā, and utkrama-jyā. The Hindus defined these as functions of an arc of a circle, not of an angle, hence their association with a bow string, and hence the "chord of an arc" for the arc is called "a bow" (dhanu, cāpa).
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According to Bartel Leendert van der Waerden, Seleucus may have constructed his heliocentric theory by determining the constants of a geometric model and by developing methods to compute planetary positions using this model, as Nicolaus Copernicus later did in the 16th century. He may have used trigonometric methods that were available in his time, as he was a contemporary of Hipparchus. Since the time of Heraclides Ponticus (387 BC-312 BC), the inferior planets Mercury and Venus have been at times named solar planets, as their positions diverge from the Sun by only a small angle. According to the Greek geographer Strabo, Seleucus was also the first to assume the universe to be infinite.
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They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry. Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya". sine rule in Yuktibhāṣā Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology. Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal place-value system first appears.
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Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular (see History of the function concept).Struik (1969), 367 In the 18th century, "function" lost these geometrical associations. Leibniz also believed that the sum of an infinite number of zeros would equal to one half using the analogy of the creation of the world from nothing. Leibniz was also one of the pioneers in actuarial science, calculating the purchase price of life annuities and the liquidation of a state's debt.
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During a summer the two students started doing their own research on a roulette wheel which they had bought. Using instruments including a camera and an oscilloscope to keep track of the motion of the roulette wheel, they eventually figured out a formula involving trigonometric functions and four variables, among them the period of rotation of the roulette wheel and the period of rotation of the ball around the roulette wheel. Since the calculations were very complicated, they decided to build a computer customized for the purpose of being fed data about the roulette wheel and the ball and to return a prediction of which of the roulette wheel's octants the ball would fall on. The computer was concealable, designed to be invisible to an onlooker.
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At the beginning of the 19th century geographers generally believed the Andes, thought to be reaching up to about , were the highest mountains in the world. Some Himalayan peaks were measured to be higher although measurements of those in Tibet and Nepal were over very great distances from Indian territory and so were known not to be very accurate. The Great Trigonometric Survey of India reached Himalaya early in the 19th century and because it was in the British Raj the Garhwal District (once the Garhwal Kingdom and after independence a division now in the state of Uttarakhand) was surveyed before the Nepalese and Tibetan regions. By 1820 the highest mountain to have been measured was Peak XIV in Garhwal Himalaya at .
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In spite of the hardships of poverty, enemy bombings, and relative academic isolation from the rest of the world, Hua continued to produce first- rate mathematics. During his eight years there, Hua studied Vinogradov's seminal method of estimating trigonometric sums and reformulated it in sharper form, in what is now known universally as Vinogradov's mean value theorem. This famous result is central to improved versions of the Hilbert–Waring theorem, and has important applications to the study of the Riemann zeta function. Hua wrote up this work in a booklet titled Additive Theory of Prime Numbers that was accepted for publication in Russia as early as 1940, but owing to the war, did not appear in expanded form until 1947 as a monograph of the Steklov Institute.
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Some sundials both decline and recline, in that their shadow- receiving plane is not oriented with a cardinal direction (such as true North or true South) and is neither horizontal nor vertical nor equatorial. For example, such a sundial might be found on a roof that was not oriented in a cardinal direction. The formulae describing the spacing of the hour-lines on such dials are rather more complicated than those for simpler dials. There are various solution approaches, including some using the methods of rotation matrices, and some making a 3D model of the reclined-declined plane and its vertical declined counterpart plane, extracting the geometrical relationships between the hour angle components on both these planes and then reducing the trigonometric algebra.
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In other words, many maps of the time were the product of an accumulation of knowledge and surveys over time, but some may be slightly inaccurate or not scientific enough. He is more concerned with the systematic measurement and calculations of distances, of maps and areas, inspiring him to attempt to improve maps from mere descriptions of general locations of areas with possible errors, to in-depth mathematical and descriptive maps that can withstand the test of time. He then goes on to describe the methods in depth for calculating the distance between two fixed points using the equators, poles, meridians, longitude, and latitude to create an imaginary triangle. Then using relatively simple trigonometric functions using angles, you can reach a mathematically deduced number.
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Nevertheless, Pytheas did obtain latitudes, which, according to Strabo, he expressed in proportions of the gnōmōn ("index"), or trigonometric tangents of angles of elevation to celestial bodies. They were measured on the gnōmōn, the vertical leg of a right triangle, and the flat leg of the triangle. The imaginary hypotenuse looked along the line of sight to the celestial body or marked the edge of a shadow cast by the vertical leg on the horizontal leg. Pytheas took the altitude of the Sun at Massalia at noon on the longest day of the year and found that the tangent was the proportion of 120 (the length of the gnōmōn) to 1/5 less than 42 (the length of the shadow).
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In response, Russell began providing various excuses as to why he could not do so. One of these was the lack of a trigonometric function routine needed to calculate the trajectories of the spacecraft. This prompted Alan Kotok of the TMRC to call DEC, who informed him that they had such a routine already written. Kotok drove to DEC to pick up a tape containing the code, slammed it down in front of Russell, and asked what other excuses he had. Russell, later explaining that "I looked around and I didn't find an excuse, so I had to settle down and do some figuring", started writing the code around the time that the PDP-1's display was installed in December 1961.
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Mics with such anvils are usually called "thread mics" or "pitch mics" (because they directly measure the pitch diameter). Users who lack thread mics rely instead on the "3-wire method", which involves placing 3 short pieces of wire (or gauge pins) of known diameter into the valleys of the thread and then measuring from wire to wire with standard (flat) anvils. A conversion factor (produced by a straightforward trigonometric calculation) is then multiplied with the measured value to infer a measurement of the thread's pitch diameter. Tables of these conversion factors were established many decades ago for all standard thread sizes, so today a user need only take the measurement and then perform the table lookup (as opposed to recalculating each time).
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The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.
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Babbage was the first to study the problem of finding a functional square root . To distinguish exponentiation from function composition, the common usage is to write the exponential exponent after the parenthesis enclosing the argument of the function; that is, means , and means . For historical reasons, and because of the ambiguity resulting of not enclosing arguments with parentheses, a superscript after a function name applied specifically to the trigonometric and hyperbolic functions has a deviating meaning: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of still denotes the inverse function. That is, is just a shorthand way to write without using parentheses, whereas refers to the inverse function of the sine, also called .
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The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha- vakhya, of unknown authorship.
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Biruni's method was intended to avoid "walking across hot, dusty deserts," and the idea came to him when he was on top of a tall mountain in India. From the top of the mountain, he sighted the angle to the horizon which, along with the mountain's height (which he calculated beforehand), allowed him to calculate the curvature of the Earth. He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.Jim Al-Khalili, , BBC According to John J. O'Connor and Edmund F. Robertson, ;Applications Muslim scholars who held to the round Earth theory used it for a quintessentially Islamic purpose: to calculate the distance and direction from any given point on the Earth to Mecca.
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The Hasselbrack, at ,Height of the hill according to the Geological State Office of Hamburg is the highest point in the state of Hamburg, Germany. It is located on the southern border of the city state with Lower Saxony in the "Black Hills" (Schwarze Berge), a northern outlier of the Harburg Hills in the quarter of Neugraben-Fischbek. It lies within the Rosengarten State Forest close to the Daerstorf Heath (Daerstorfer Heide) between the settlement of Waldfrieden in the north (which belongs to Fischbek), Neu Wulmstorf-Tempelberg in the west and Rosengarten-Alvesen in the east. On the "summit" of the Hasselbrack there is a trigonometric point, that is located in the wood just a few metres from the footpath and which marks the highest point.
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Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation. The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook Spherical trigonometry for the use of colleges and Schools.
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CORDIC (for COordinate Rotation DIgital Computer), also known as Volder's algorithm, including Circular CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, and exponentials and logarithms with arbitrary base, typically converging with one digit (or bit) per iteration. CORDIC is therefore also an example of digit-by-digit algorithms. CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and FPGAs), as the only operations it requires are additions, subtractions, bitshift and lookup tables.
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Heliotropes were used in surveys from Gauss's survey in Germany in 1821 through the late 1980s, when GPS measurements replaced the use of the heliotrope in long distance surveys. Colonel Sir George Everest introduced the use of heliotropes into the Great Trigonometric Survey in India around 1831, and the US Coast and Geographic Survey used heliotropes to survey the United States. The Indian specification for heliotropes was updated in 1981, and the American military specification for heliotropes (MIL-H-20194E) was retired on 8 December 1995. Surveyors used the heliotrope as a specialized form of survey target; it was employed during large triangulation surveys where, because of the great distance between stations (usually twenty miles or more), a regular target would be indistinct or invisible.
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In the late 11th century, Omar Khayyám (1048-1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, Nasīr al-Dīn al- Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. Nasir al-Din al- Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.
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Important in fields such as surveying, railway engineering (for example to lay out railroad curves and superelevation), civil engineering, astronomy, and spherical trigonometry up into the 1980s, the exsecant function is now little-used. Mainly, this is because the broad availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. The reason to define a special function for the exsecant is similar to the rationale for the versine: for small angles θ, the sec(θ) function approaches one, and so using the above formula for the exsecant will involve the subtraction of two nearly equal quantities, resulting in catastrophic cancellation. Thus, a table of the secant function would need a very high accuracy to be used for the exsecant, making a specialized exsecant table useful.
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Reprinted 1987 by Diadem Books, Graham's confusion was partly due to the poor quality of the maps of the area, and on his return to civilisation he was critical of the Great Trigonometric Survey, suggesting that its surveyors should be trained in mountaineering by the Swiss Army, whom he credited with the finest cartographic work in the world at the time. The criticism was not well received by the Survey, and it may have made Graham more enemies to cast doubt on his accomplishments. Kabru, which Graham claimed to have climbed. After the Garhwal trip, Graham and his companions returned to the Kanchenjunga area for the climax of their campaign; an attempt on Kabru, which Graham claimed to have climbed by the East Face in three days, reaching the summit on 8 September.
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The HP-9100 series was built entirely from discrete transistor logic with no integrated circuits, and was one of the first uses of the CORDIC algorithm for trigonometric computation in a personal computing device, as well as the first calculator based on reverse Polish notation (RPN) entry. HP became closely identified with RPN calculators from then on, and even today some of their high-end calculators (particularly the long-lived HP-12C financial calculator and the HP-48 series of graphing calculators) still offer RPN as their default input mode due to having garnered a very large following. The HP-35, introduced on February 1, 1972, was Hewlett-Packard's first pocket calculator and the world's first handheld scientific calculator.HP-35 Scientific Calculator Awarded IEEE Milestone Like some of HP's desktop calculators it used RPN.
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This is in contrast to older cadastral surveys, which primarily show property and governmental boundaries. The first multi-sheet topographic map series of an entire country, the Carte géométrique de la France, was completed in 1789.Library of Congress, Geography and Maps: General Collections The Great Trigonometric Survey of India, started by the East India Company in 1802, then taken over by the British Raj after 1857 was notable as a successful effort on a larger scale and for accurately determining heights of Himalayan peaks from viewpoints over one hundred miles distant. Global indexing system first developed for International Map of the World Topographic surveys were prepared by the military to assist in planning for battle and for defensive emplacements (thus the name and history of the United Kingdom's Ordnance Survey).
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The Intel 8231 (and revised 8231A) is the Arithmetic Processing Unit (APU). It offered 32-bit "double" precision (a term later and more commonly used to describe 64-bit floating-point numbers, whilst 32-bit is considered "single" precision) floating-point, and 16-bit or 32-bit ("single" or "double" precision) fixed-point calculation of 14 different arithmetic and trigonometric functions to a proprietary standard. The APU used the Chebyshev polynomials using the algorithms provided here. The available APU version of 4-MHz was for USD $235.00 and 2-MHz was for USD $149.00 in quantities of 100 or more.Intel Corporation, "Intel peripherals enhance 8086 system design", Intel Preview Special Issue: 16-Bit Solution, May/June 1980, Pg. 22 The later Intel 8232 is the Floating Point Processor Unit (FPU).
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Using the FEE, it is possible to prove the following theorem: Theorem: Let y=f(x) be an elementary transcendental function, that is the exponential function, or a trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the inverses. Then : s_f(n) = O(M(n)\log^2n). \, Here s_f(n) is the complexity of computation (bit) of the function f(x) with accuracy up to n digits, M(n) is the complexity of multiplication of two n-digit integers. The algorithms based on the method FEE include the algorithms for fast calculation of any elementary transcendental function for any value of the argument, the classical constants e, \pi, the Euler constant \gamma, the Catalan and the Apéry constants,Karatsuba E. A., Fast evaluation of \zeta(3), Probl.
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The methods of analysis had been planned in outline by William Yolland, his predecessor at the head of the Trigonometric Section, but it fell to Clarke to finalize the methods and carry them through to completion. This he achieved in the four years from 1854 to 1858: the report was published as but it is entirely Clarke's work. The basic data was the collection of angle bearings taken from each of the 289 stations towards a number of other stations, typically from three to ten in number. The multiple observations were first subjected to a least squares error analysis to extract the most likely angles and then the triangles formed by the corrected bearings were adjusted simultaneously, again by least squares methods, to find the most likely geometry for the whole mesh.
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Felix Candela was born in Madrid, Spain in 1910. In 1927 Candela enrolled in La Escuela Superior de Arquitectura (Madrid Superior Technical School of Architecture), graduating in 1935; at which time Candela traveled to Germany to further study architecture. Early after he started classes, he developed a very keen sense of geometry and started teaching other students in private lessons. In his junior year, his visual intelligence and his descriptive geometric and trigonometric talent helped him catch the eye of Luis Vegas. Vegas was his material strength professor, and gave Candela the honorary title of “Luis Vegas’ Helper”. While “helping” Vegas, Candela entered many architecture competitions and won most of them. Unlike many of his peers, Candela didn’t show intellectual or aesthetic efforts in school. He didn’t even like pure mathematics.
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Yuktibhāṣā (), also known as ' (Compendium of Astronomical Rationale), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. The work was unique for its time, since it contained proofs and derivations of the theorems that it presented; something unusual for Indian mathematicians of that era. Some of its important topics include the infinite series expansions of functions; power series, including of π and π/4; trigonometric series of sine, cosine, tangent and arctangent; Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences; and tests of convergence.
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On January 12, 2018, Team Z premiered the "Trigonometric Function"(Chinese:三角函数) performance. On March 28, in conjunction with the SNH48 Group, the 6th EP "抱紧处理" was issued. On April 2, the Wolf Man killed the marathon in the SNH48 Group's exclusive Werewolf app "48 Werewolf Kill"(Chinese:48狼人杀). From April 6 to 8, the GNZ48 debut two-year anniversary photo album "你所不知道的…" is also the first GNZ48 photo book. From April 27 to 30, together with BEJ48 in the SNH48 Group's exclusive werewolf killing APP "48 Werewolf Kill"(Chinese:48狼人杀) held the "悠点泰度" large werewolf marathon competition, more than 100 members took turns in 48 days of marathon in 4 days Live match.
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This is comparable to the interface provided by the more recent TI-82, TI-83, and so on. This system is capable of such tasks as two-dimensional parametric graphing (in addition to standard two-dimensional function graphing), trigonometric calculations in units of either degrees or radians, simple drawing capabilities, creation and manipulation of matrices up to 6x6 in size, and programming in Texas Instruments' native TI-BASIC programming language. In late 2009 an exploit was found that can be used to execute machine code on the TI-81, using manual input of code instead of sending programs using a link cable. As with its successors, the TI-81 is powered by four AAA batteries and one CR1616 or CR1620 lithium backup battery (to ensure programs are kept when the AAA batteries are being changed).
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In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z−w), taking the limit value at w, is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.
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All methods of field survey were used, much based on the urgency of the need: astronomy, triangulation, theodolite and chain traverse, plane-tabling, compass traverse, levelling, trigonometric heighting, barometry, intersection of prominent features behind the enemy to improve artillery targeting. A narrow triangulation chain was carried from the astronomic datum at Wau, north through Bulolo into the Markham Valley in anticipation of the Australian attacks on Japanese held Lae. In October the sub-section was to be relieved and they walked out 110km to the south coast on the Wau-Bulldog road and then by steamer to Port Moresby. After sixteen months of active service 8 Aust Fd Svy Sect was absorbed as No 2 Field Section of 6th Aust Army Topographic Survey Company and moved to Australia.
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Pitiscus supported Frederick's subsequent measures against the Roman Catholic Church. Pitiscus achieved fame with his influential work written in Latin, called Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus (1595, first edition printed in Heidelberg), which introducedGroundbreaking Scientific Experiments, Inventions, and Discoveries the word trigonometry to the English and French languages, translations into which had appeared in 1614 and 1619, respectively. It consists of five books on plane and spherical trigonometry. Pitiscus is sometimes credited with inventing the decimal point, the symbol separating integers from decimal fractions, which appears in his trigonometrical tables and was subsequently accepted by John Napier in his logarithmic papers (1614 and 1619). Pitiscus edited Thesaurus mathematicus (1613) in which he improved the trigonometric tables of Georg Joachim Rheticus and also corrected Rheticus’s Magnus Canon doctrinæ triangulorum.
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Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair of rational numbers :(x,y) and a line :ax + by + c = 0, as a general linear equation with rational coefficients , and . By avoiding calculations that rely on square root operations giving only approximate distances between points, or standard trigonometric functions (and their inverses), giving only truncated polynomial approximations of angles (or their projections) geometry becomes entirely algebraic. There is no assumption, in other words, of the existence of real number solutions to problems, with results instead given over the field of rational numbers, their algebraic field extensions, or finite fields. Following this, it is claimed, makes many classical results of Euclidean geometry applicable in rational form (as quadratic analogs) over any field not of characteristic two.
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In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner. Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al- Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī. During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.
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Scientific calculators are used widely in situations that require quick access to certain mathematical functions, especially those that were once looked up in mathematical tables, such as trigonometric functions or logarithms. They are also used for calculations of very large or very small numbers, as in some aspects of astronomy, physics, and chemistry. They are very often required for math classes from the junior high school level through college, and are generally either permitted or required on many standardized tests covering math and science subjects; as a result, many are sold into educational markets to cover this demand, and some high-end models include features making it easier to translate a problem on a textbook page into calculator input, e.g. by providing a method to enter an entire problem in as it is written on the page using simple formatting tools.
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Mathematician and astronomer Minggatu of Sharaid discovered nine trigonometric equations and wrote 42 volumes of "The Roots of Regularites" (Зvй тогтлын бvрэн эх сурвалж), 5 volumes in linguistics (дуун ухаан), and 53 volumes of work on mathematics. In the area of historiography and literature, the Shira Tuuji was written in the 16th century, the Altan Tobchi of Lubsandanzan was written in the first half of the 17th century, and the Erdeniin Tobchi of Sagan Secen Hongtaiji (a descendant of Hutuhtai Secen Hongtaiji), was written in 1662. In the 1620s, Tsogtu Hongtaiji of Khalkha wrote his famous philosophic poems and Legdan Hutuhtu Khan had the 108 volumes of Kangyur and 225 volumes of Tengyur translated into the Mongolian language. A translation theory work, The Source of Wisdom (Мэргэд гарахын орон) was written under leadership of Rolbiidorji, Janjaa Hutuhtu II.
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The notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham in works such as Uniplanar Algebra (1893), or by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898). It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. It is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms, or when exponential functions shouldn't be used for some reason in math education. In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL)), available for many compilers, programming languages (including C, C++, Common Lisp, D, Fortran, Haskell, Julia), and operating systems (including Windows, Linux, macOS and HP-UX).
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Michael Hennessy, one of Waugh's assistants, had begun designating peaks based on Roman numerals, with Kangchenjunga named Peak IX. Peak "b" now became known as Peak XV. In 1852, stationed at the survey headquarters in Dehradun, Radhanath Sikdar, an Indian mathematician and surveyor from Bengal was the first to identify Everest as the world's highest peak, using trigonometric calculations based on Nicolson's measurements. An official announcement that Peak XV was the highest was delayed for several years as the calculations were repeatedly verified. Waugh began work on Nicolson's data in 1854, and along with his staff spent almost two years working on the numbers, having to deal with the problems of light refraction, barometric pressure, and temperature over the vast distances of the observations. Finally, in March 1856 he announced his findings in a letter to his deputy in Calcutta.
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In mathematics, an addition theorem is a formula such as that for the exponential function :ex + y = ex·ey that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is more apparent than real, in that case, since there cos is an algebraic function of sin (in other words, we usually take their functions both as defined on the unit circle). The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions. To 'classify' addition theorems it is necessary to put some restriction on the type of function G admitted, such that :F(x + y) = G(F(x), F(y)).
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The line element given above, with f,g, regarded as undetermined functions of the isotropic coordinate r, is often used as a metric Ansatz in deriving static spherically symmetric solutions in general relativity (or other metric theories of gravitation). As an illustration, we will sketch how to compute the connection and curvature using Cartan's exterior calculus method. First, we read off the line element a coframe field, : \sigma^0 = -a(r) \, dt : \sigma^1 = b(r) \, dr : \sigma^2 = b(r) \, r \, d\theta : \sigma^3 = b(r) \, r \, \sin(\theta) \, d\phi where we regard a, \,b as undetermined smooth functions of r. (The fact that our spacetime admits a frame having this particular trigonometric form is yet another equivalent expression of the notion of an isotropic chart in a static, spherically symmetric Lorentzian manifold).
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He suggested that the correct values might be found by combining the Paris Observatory figures with a precise trigonometric survey between the two observatories. (The French surveys had already been carried out in the course of the preparation of the map of France.) This criticism was roundly rejected by Nevil Maskelyne who was convinced of the accuracy of the Greenwich measurements but, at the same time, he realised that Cassini's memoir provided a means of promoting government funding for a survey which would be valuable in its own right. Approval was granted and Sir Joseph Banks, president of the Royal Society, proposed that Roy should lead the project. Roy gladly accepted and set matters in motion by submitting to the Crown a grossly-underestimated budget for manpower (by far the largest element) and new precision instruments to be constructed by Jesse Ramsden.
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Each of these steps may be performed with simple trigonometric calculations, and as Collins and Stephenson argue, the system of radii converges rapidly to a unique fixed point for which all covering angles are exactly 2π. Once the system has converged, the circles may be placed one at a time, at each step using the positions and radii of two neighboring circles to determine the center of each successive circle. describes a similar iterative technique for finding simultaneous packings of a polyhedral graph and its dual, in which the dual circles are at right angles to the primal circles. He proves that the method takes time polynomial in the number of circles and in log 1/ε, where ε is a bound on the distance of the centers and radii of the computed packing from those in an optimal packing.
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About 1835, Jerrard demonstrated that quintics can be solved by using ultraradicals (also known as Bring radicals), the unique real root of for real numbers . In 1858 Charles Hermite showed that the Bring radical could be characterized in terms of the Jacobi theta functions and their associated elliptic modular functions, using an approach similar to the more familiar approach of solving cubic equations by means of trigonometric functions. At around the same time, Leopold Kronecker, using group theory, developed a simpler way of deriving Hermite's result, as had Francesco Brioschi. Later, Felix Klein came up with a method that relates the symmetries of the icosahedron, Galois theory, and the elliptic modular functions that are featured in Hermite's solution, giving an explanation for why they should appear at all, and developed his own solution in terms of generalized hypergeometric functions.
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The fundamental difference between a calculator and computer is that a computer can be programmed in a way that allows the program to take different branches according to intermediate results, while calculators are pre-designed with specific functions (such as addition, multiplication, and logarithms) built in. The distinction is not clear-cut: some devices classed as programmable calculators have programming functions, sometimes with support for programming languages (such as RPL or TI-BASIC). For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in read-only memory (ROM), and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require much multiplication. Bit serial logic designs are more common in calculators whereas bit parallel designs dominate general-purpose computers, because a bit serial design minimizes chip complexity, but takes many more clock cycles.
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LP 876-10 is also associated with the Fomalhaut system, making it a trinary star. In October 2013, Eric Mamajek and collaborators from the RECONS consortium announced that the previously known high-proper-motion star LP 876-10 had a distance, velocity, and color-magnitude position consistent with being another member of the Fomalhaut system. LP 876-10 was originally catalogued as a high-proper-motion star by Willem Luyten in his 1979 NLTT catalogue; however, a precise trigonometric parallax and radial velocity was only measured quite recently. LP 876-10 is a red dwarf of spectral type M4V, and located even further from Fomalhaut A than TW PsA—about 5.7° away from Fomalhaut A in the sky, in the neighbouring constellation Aquarius, whereas both Fomalhaut A and TW PsA are located in constellation Piscis Austrinus.
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Tide Predicting Machine No. 2 is based on the first accurate mathematical approach for predicting tides, which was developed around 1867 by Sir William Thomson (who later became Lord Kelvin) and later refined by Sir George Darwin. This approach, called “harmonic analysis,” approximates tide heights by a summation of cosine terms, each of which has a different frequency. The formula for sea height is represented as : H_0 + A_1 \cos(\omega_1 t+\varphi_1)+A_2 \cos(\omega_2 t+\varphi_2)+A_3 \cos(\omega_3 t+\varphi_3)+\cdots containing 10, 20 or even more trigonometric terms. H_0 is the height of mean sea level. For each term i = 1, \ldots, n, A_i is the amplitude of the term’s contribution to tide height above mean sea level, \omega_i determines the frequency of the term, t is the time, and \phi_i is the relative phase of the term.
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Louis Colas (1869-1929) was so convinced that Charlemagne had planted the cross atop the summit of Orzanzurieta, the highest peak in the region, when he repaired the road to Zaragoza, then he climbed the Pyrenees in search of its remains in 1910. The remains of a cross he did report, but these were shown to be the remains of a trigonometric column used by the military.José María Lacarra (1971), Estudios de Historia Navarra: Navarra (Ediciones y Libros), 109. Arturo Campión (1854-1937) wrote of a visit he made to Orzanzurieta in the 1880s: "...antes de que la columna se arruinase, estuve sentado durante largo tiempo en los escalones de piedra, contemplando el admirable panorama que desde allí se descubre" (before the column went to ruin, I sat for a long time on the stone steps, contemplating the admirable panorama found from there).
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A spherical triangle In sixteenth century Europe, celestial navigation of ships on long voyages relied heavily on ephemerides to determine their position and course. These voluminous charts prepared by astronomers detailed the position of stars and planets at various points in time. The models used to compute these were based on spherical trigonometry, which relates the angles and arc lengths of spherical triangles (see diagram, right) using formulas such as: :\cos a = \cos b \cos c + \sin b \sin c \cos \alpha and :\sin b \sin \alpha = \sin a \sin \beta where a, b and c are the angles subtended at the centre of the sphere by the corresponding arcs. When one quantity in such a formula is unknown but the others are known, the unknown quantity can be computed using a series of multiplications, divisions, and trigonometric table lookups.
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Sinclair's brief to the pair was fairly non-specific but primarily concerned remedying a key defect of the ZX80 so that the new machine could be used for practical programming and calculations. Vickers later recalled: The new ROM incorporated trigonometric and floating point functions, which its predecessor had lacked – the ZX80 could only deal with whole numbers. Grant came up with one of the ZX81's more novel features, a syntax checker that indicated errors in BASIC code as soon as it was entered (rather than, as was standard at the time, only disclosing coding errors when a program was run). Unfortunately for Vickers, he introduced a briefly notorious error – the so-called "square-root bug" that caused the square root of 0.25 to be returned erroneously as 1.3591409 – as a result of problems with integrating the ZX Printer code into the ROM.
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From 2001-2010, ELM research mainly focused on the unified learning framework for "generalized" single-hidden layer feedforward neural networks (SLFNs), including but not limited to sigmoid networks, RBF networks, threshold networks, trigonometric networks, fuzzy inference systems, Fourier series, Laplacian transform, wavelet networks, etc. One significant achievement made in those years is to successfully prove the universal approximation and classification capabilities of ELM in theory. From 2010 to 2015, ELM research extended to the unified learning framework for kernel learning, SVM and a few typical feature learning methods such as Principal Component Analysis (PCA) and Non-negative Matrix Factorization (NMF). It is shown that SVM actually provides suboptimal solutions compared to ELM, and ELM can provide the whitebox kernel mapping, which is implemented by ELM random feature mapping, instead of the blackbox kernel used in SVM.
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Discrete Fourier transforms are often used to solve partial differential equations, where again the DFT is used as an approximation for the Fourier series (which is recovered in the limit of infinite N). The advantage of this approach is that it expands the signal in complex exponentials einx, which are eigenfunctions of differentiation: d/dx einx = in einx. Thus, in the Fourier representation, differentiation is simple—we just multiply by i n. (Note, however, that the choice of n is not unique due to aliasing; for the method to be convergent, a choice similar to that in the trigonometric interpolation section above should be used.) A linear differential equation with constant coefficients is transformed into an easily solvable algebraic equation. One then uses the inverse DFT to transform the result back into the ordinary spatial representation.
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The hill castle on the top of the drumlin from northwest The Waldburg is a Hilltop castle located on a natural elevation, a drumlin from the last glacial period, at height above sea level. The raised situation with view (when suitable weather conditions) to the west up to the Hohentwiel near Singen, to the north up to the Ulm Minster, to the east far back in the Alpine foothills and southwards far into the Swiss Alps and the Lake Constance made the Waldburg to an important trigonometric point also for land surveying in the early 19th century of the ordnance survey. The steepen drumlin already offers by his very big slope angle an almost ideal military protection for a castle construction, however, complicated also the building and expansion more than seven centuries considerably. The hill castle was very woody till the eighties of the 20th century.
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The scholar-cleric Ramon Llull of Majorca, was the first writer to refer to a rule to solve the traverse problem of navigation. In his Arbor Scientiae (1295), in the section of questions on geometry, Llul writes: What Llull seems to be trying to explain is that a ship actually sailing E, but intending to sail SE, it can figure out how much of its intended southeastward distance it has already made good – what Italians called the "avanzar", but Lull seems to call the "miliaria in mari". Llull does not explain exactly how, but refers only to an "instrument", presumably some sort of trigonometric table. Lull is implying that mariners can calculate the miliaria on the intended course by multiplying the distance actually sailed on the erroneous course by the cosine of the angle between the two routes.This interpretation is originally due to Taylor (1956: pp. 117–19).
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By 1941, prompted by the British experience, Americans began to understand the need for a much expanded air reconnaissance concept. The F-series, which denoted photographic reconnaissance, was then led by the F-3A, a modified A-20 Havoc. Thanks in large part to the advocacy of the Director of Photographic Intelligence, the also very controversial Colonel Minton Kaye, a run of 100 Lockheed P-38s were set aside for modification to F-4 standard, incorporating the trigonometric mount that both Kaye and Cotton had pioneered prior to the war. Despite the promising performance of the F-4, there were so many technical problems with the early versions that the model was largely rejected by its crews when it did reach combat zones. The RAF rejected the P-38, as well.Goddard, 297, 299 The first U.S. operational reconnaissance experience was gained in the Australian theater.
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Before the advent of computers, printed lookup tables of values were used by people to speed up hand calculations of complex functions, such as in trigonometric tables, logarithm tables, and tables of statistical density functions School children are often taught to memorize "times tables" to avoid calculations of the most commonly used numbers (up to 9 x 9 or 12 x 12). Even as early as 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" Maher, David. W. J. and John F. Makowski. "Literary Evidence for Roman Arithmetic With Fractions", 'Classical Philology' (2001) Vol.
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Rheticus returned to the University of Wittenberg in October 1541, then elected dean of the Faculty of Arts as well as joining the theological faculty. In May 1542, he traveled to Nürnberg to supervise the printing by Johannes Petreius of the first edition of De revolutionibus in which he included tables of trigonometric functions he had calculated in further support of Copernicus' work, but had to leave in fall to take a position at the University of Leipzig, and Andreas Osiander replaced him. A theologian, Osiander would use this role to add an unauthorized preface in a would-be attempt to avoid censorship, explicitly describing the theory discussed therein as a model of pure hypothesis predicated on assumptions that are coincidentally consistent with the calculations.Hooykas, 1984 Towards this, Rheticus would allegedly deface every such copy he came across. Copernicus' major work would eventually be published shortly before his death in 1543.
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The design of a simple Mariner's Quadrant Another latitude-divining instrument was the quadrant. Although their functions were the same, they were developed independently for separate use; the kamal was nautical while the quadrant was astronomical. The idea was born with Ptolemy in the 1st century, and was once again pioneered by the Arabs; its intended purpose was to solve trigonometric functions dictating celestial movement. However, the dual sighting mechanism of the period was too complicated, time consuming, and required too many men for practical use aboard a ship, as a result Iberian captains stripped it down to its most basic parts in order for it to be applied to navigation.ThinkQuest: Library, “Early Navigational Instruments ” The apparatus itself was either a wood or brass quarter circle with degree measurements painted along its length and a length of string attached to the point to serve as a plumb bob.
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Presanella may have first been climbed by surveyors in 1854. Eduard Pechmann's 1865 Notizen zur Höhen- und Profilkarte has Presanella's height with two digits precision (1878.26 Viennese Klaster or 3,562.1 m), which in this list indicated that a measurement was taken from the summit during the trigonometric survey, which for Presanella was done in 1854. This possible ascent is otherwise unrecorded.Karl Schulz, Die Adamello Gruppe in Die Erschliessung der Ostalpen, Volume 2, Deutscher und Österreichischer Alpenverein, 1894, pp. 234-237. In 1862, the Viennese jurist :de:Anton von Ruthner and the guides Kuenz from Martell and DelperoNamed Bortolo or Bortolameo Delpero in some sources Schulz called him “an Italian from Vermiglio (Del Pero?)” and Freshfield doesn’t give a name, but mentions that they hired a "porter" from Vermiglio who had guided a “German professor” in his attempt two years earlier and could lead them to the foot of the mountain.
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The poetry of O Fortuna was actually the work of itinerant goliards, found in the German Benedictine monastery of Benediktbeuern Abbey. The hoax was lent an air of credibility because often medieval monks did discover scientific and mathematical theories, only to have them hidden or shelved due to persecution or simply ignored because publication prior to the invention of the printing press was difficult at best. Mr. Girvan adds to this suggestion by associating Udo with several other more legitimate discoveries where an author was considered ahead of his time in terms of a scientific theory of some sort that is now established as a mainstream theory but was considered fringe science at the time. Another aspect of the deception was that it was very common for pre-20th century mathematicians to spend incredible amounts of time on hand calculations such as a logarithm table or trigonometric functions.
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A portion of the qibla table compiled by astronomer and alt=A table written in Arabic The study of astronomy—known as ilm al-falak () in the Islamic intellectual tradition—began to appear in the Islamic World in the second half of the 8th century, centred in Baghdad, the principal city of the Abbasid Caliphate. Initially, the science was introduced through the works of Indian authors, but after the 9th century the works of Greek astronomers such as Ptolemy were translated into Arabic and became the main references in the field. Muslim astronomers preferred Greek astronomy because they considered it to be better supported by theoretical explanations and therefore it could be developed as an exact science; however, the influence of Indian astronomy survives especially in the compilation of astronomical tables. This new science was applied to develop new methods of determining the qibla, making use of the concept of latitude and longitude taken from Ptolemy's Geography as well as trigonometric formulae developed by Muslim scholars.
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Maoile Lunndaidh is a large and remote mountain, covering an area in excess of 10 square miles (2,590 hectares). Its location is almost equidistant from the valleys of Strathconon, Strathfarrar and Glen Carron, the nearest public road being over 10 km away. The mountain is listed in the current edition of Munro's Tables as 1007 metres (3304 feet). This height dates from the introduction of the 1:50k OS map series in the 1970s. However the current 1:25000 map shows a spot height of 1,005 metres, in better agreement with the trigonometric height of 3293.8 feet on old 6 inch maps (equivalent to 1004.1 metres after conversion to the Newlyn datum). Maoile Lunndaidh has been described as “the flattest of bulks”."Hamish‘s Mountain Walk" Page 274 Gives quote: “Flattest of bulks”. Its extensive level summit plateau is reminiscent of the Cairngorm and seems out of place amongst the west coast peaks.
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But for all of its prominence, there is as yet no understanding of what the OWL is or how it came to be. The OWL piques the interest of geologically minded persons in part because its characteristic NW-SE angle of orientation - approximately 50 to 60 degrees west of north (a little short of northwest)Estimating the northing and westing from a map and applying the usual trigonometric methods gives an angle of 59 degrees west of north (N59W, azimuth 301°) from Wallula Gap to Cape Flattery. There is a bit of a bend east of Port Angeles - the shore line between Pillar Point to Slip Point has a more westerly angle of 65 degrees - but that section is so short that the angle from Wallula Gap to Port Angeles is still 57 degrees. A line run from the strong relief at Gold Creek to the mouth of Liberty Bay and beyond - a line that runs along several seeming OWL features - has an angle of 52 deg.
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The highest peak of Toubkal The first recorded ascent was on 12 June 1923 by the Marquis de Segonzac, Vincent Berger and Hubert Dolbeau, but the mountain may well have been climbed before that date.Robin G. Collomb, Atlas Mountains, Goring: West Col, 1980 Toubkal's height was measured the following year, and determined as being Nowadays measured at 4,167 metres, the summit is crowned with a large pyramidal metal trigonometric marker, and offers views taking in most of the Atlas and Little Atlas Mountains. It is possible to climb mountain Toubkal in two days - first day up to the refuge (around seven hours), second day to the summit (around four hours ascent, three hours descent) and back to Imlil (up to five hours). In summer the mountains can be very dry, but are sometimes subject to storms. Although the temperature should remain above zero during the day, freezing conditions are possible over 3,500m.
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The Centaur upper stage was first designed and developed for launching the Surveyor lunar landers, beginning in 1966, to augment the delta-V of the Atlas rockets and give them enough payload capability to deliver the required mass of the Surveyors to the Moon. More than 100 Convair- produced Atlas-Centaur rockets (including those with their successor designations) were used to successfully launch over 100 satellites, and among their many other outer-space missions, they launched the Pioneer 10 and Pioneer 11 space probes, the first two to be launched on trajectories that carried them out of the Solar System. In addition to aircraft, missiles, and space vehicles, Convair developed the large Charactron vacuum tubes, a form of cathode ray tube (CRT) computer display with a shaped mask to form characters, and to give an example of a minor product, the CORDIC algorithms, which is widely used today to calculate trigonometric functions in calculators, field- programmable gate arrays, and other small electronic systems.
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Shahrani, M. Nazif. (1979) The Kirghiz and Wakhi of Afghanistan: Adaptation to Closed Frontiers and War University of Washington Press, Seattle, ; 1st paperback edition with new preface and epilogue (2002), p.27 In 1868, a pundit known as the Mirza, working for the Great Trigonometric Survey of India, crossed the pass.Shahrani, M. Nazif. (1979 and 2002) p.31 There were further crossings in 1874 by Captain T.E. Gordon of the British Army,Keay, J. (1983) When Men and Mountains Meet p. 256-7 in 1891 by Francis Younghusband,Younghusband, F. (1896, republished 2000) The Heart of a Continent and in 1894 by Lord Curzon.Geographical Journal (July to September 1896) cited in Mock and O'Neil 2004 Shipton Tilman Grant Application In May 1906, Sir Aurel Stein crossed the pass and reported that at that time, the pass was used by only 100 pony loads of goods each way annually.Shahrani, M. Nazif (1979 and 2002) p.
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Ibn Yunus' most famous work in Islamic astronomy, al-Zij al- Kabir al-Hakimi (c. 1000), was a handbook of astronomical tables which contained very accurate observations, many of which may have been obtained with very large astronomical instruments. According to N. M. Swerdlow, the Zij al-Kabir al-Hakimi is "a work of outstanding originality of which just over half survives".N. M. Swerdlow (1993), "Montucla's Legacy: The History of the Exact Sciences", Journal of the History of Ideas 54 (2): 299–328 [320]. Yunus expressed the solutions in his zij without mathematical symbols,Complete Dictionary of Scientific Biography, 2008 but Delambre noted in his 1819 translation of the Hakemite tables that two of Ibn Yunus' methods for determining the time from solar or stellar altitude were equivalent to the trigonometric identity 2\cos(a)\cos(b) = \cos(a+b)+\cos(a-b)David A. King, 'Islamic Math and Science', Journal for the History of Astronomy, Vol.
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Ilyin is recognized for his outstanding scientific achievements in the theory of boundary value and mixed problems for equations of mathematical physics in domains with non-smooth boundaries and discontinuous coefficients. His results for hyperbolic equations (combined with earlier results obtained by Andrey Tikhonov, O.A. Oleinik, and G. Tautz for parabolic and elliptic equations) demonstrated that in terms of domain boundary conditions the solvability of all the three problems reduces to the solvability of a simplest problem of mathematical physics, the Dirichlet problem for the Laplace equation. In the late 1960s Ilyin developed a universal method that made it possible for an arbitrary selfadjoint second-order operator in an arbitrary (not necessarily bounded) domain to establish the final conditions of uniform (on any compact) convergence for both spectral expansions themselves and their Riesz means in each of the classes of functions (Nikolsky, Sobolev-Liouville, Besov and Sigmund-Holder function classes). These conditions also proved to be novel and final for expansions into both the multiple Fourier integral and the trigonometric Fourier series.
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In 1997 Ruiz et al. had made two estimates of distance of Kelu-1—from its proper motion, assuming its observed motion is due to Solar System's motion only (about 12 parsecs), and from its apparent magnitude in J band, assuming it is same with that of GD 165 B—another L-type brown dwarf with similar spectral properties, discovered in 1988 in the system of white dwarf GD 165 (about 10 parsecs). But in 1999 preliminary trigonometric parallax of Kelu-1, measured under USNO faint-star parallax program, was obtained, and it turned out that it is located further—at about 19 parsecs, and so it is more luminous than GD 165 B. There were two possible explanations of overluminosity of Kelu-1: it is either young (age less than 0.1 Gyr) or binary. However, observations of Kelu-1 with near-infrared camera NICMOS on Hubble Space Telescope carried out on 1998 August 14, did not reveal the presence of any companion with separation greater than 300 mas and magnitude difference less than 6.7 mag.
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Broken lances lying along perspective lines in Paolo Uccello's The Battle of San Romano, 1438 Small stellated dodecahedron, from De divina proportione by Luca Pacioli, woodcut by Leonardo da Vinci. Venice, 1509 Albrecht Dürer's 1514 engraving Melencolia, with a truncated triangular trapezohedron and a magic square Rencontre dans la porte tournante by Man Ray, 1922, with helix Four- dimensional geometry in Painting 2006-7 by Tony Robbin Quintrino by Bathsheba Grossman, 2007, a sculpture with dodecahedral symmetry Heart by Hamid Naderi Yeganeh, 2014, using a family of trigonometric equations This is a list of artists who actively explored mathematics in their artworks. Art forms practised by these artists include painting, sculpture, architecture, textiles and origami. Some artists such as Piero della Francesca and Luca Pacioli went so far as to write books on mathematics in art. Della Francesca wrote books on solid geometry and the emerging field of perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De corporibus regularibus (Regular Solids),Piero della Francesca, De Prospectiva Pingendi, ed.
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Sine, cosine, and versine of angle θ in terms of a unit circle with radius 1, centered at O. This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow. If the arc ADB of the double-angle Δ = 2θ is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft". Graphs of historical trigonometric functions compared with sin and cos - in [ the SVG file,] hover over or click a graph to highlight it The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine (sinus versus). The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle: For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) is the distance AC (half of the chord).
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Diagram explaining the meaning of the values in Madhava's table To understand the meaning of the values tabulated by Madhava, consider some angle whose measure is A. Consider a circle of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the perpendicular QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles \anglePOS in arcminutes, arcseconds and sixtieths of an arcsecond. As an example, let A be an angle whose measure is 22.50°. In Madhava's table, the entry corresponding to 22.50° is the measure in arcminutes, arcseconds and sixtieths of arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and, :0.382683432363 radians = 180 / π × 0.382683432363 degrees = 21.926145564094 degrees.
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The embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations. The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle s by s = c + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc.Katz, 308. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).. As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.
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Operation Substitution is one of the operation reduction techniques where certain costly operations are substituted by relatively cheaper operations which reduce power consumption. Some typical examples of operation substitution techniques are given as follows: #Multiplication by Adds/Subtracts: The multiplication of two numbers if costly compared to addition of two numbers therefore substituting it with addition is profitable. For example, to calculate y = x2 \+ Ax + B we can calculate x2, Ax, and add both of them to B which has 2 multiplications, 3 additions or we can convert it into y = x(x+A) + B where we can calculate x+A multiply it with x and add B where we have 1 multiplication and 2 additions, both approaches have same critical path length but 2nd one has lesser multiplications which saves power. #Computation of Sine/cosine/tan: Computing trigonometric functions might also turn out to be quite costly where as substituting them with lesser order Taylor expansion makes them less power consuming but we may lose on approximation grounds which is a trade-off one should keep in mind.
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Thanks to improvements in instruments and accuracy, Picard's is rated as the first reasonably accurate measurement of the radius of the earth. Over the next century this work was extended most notably by the Cassini family: between 1683 and 1718 Jean-Dominique Cassini and his son Jacques Cassini surveyed the whole of the Paris meridian from Dunkirk to Perpignan; and between 1733 and 1740 Jacques and his son César Cassini undertook the first triangulation of the whole country, including a re-surveying of the meridian arc, leading to the publication in 1745 of the first map of France constructed on rigorous principles. Triangulation methods were by now well established for local mapmaking, but it was only towards the end of the 18th century that other countries began to establish detailed triangulation network surveys to map whole countries. The Principal Triangulation of Great Britain was begun by the Ordnance Survey in 1783, though not completed until 1853; and the Great Trigonometric Survey of India, which ultimately named and mapped Mount Everest and the other Himalayan peaks, was begun in 1801.
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The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam. Oxford University Press. p. 321. the discovery of the pulmonary circulation by Ibn al-Nafis, the discovery of the itch mite parasite by Ibn Zuhr, the first use of irrational numbers as an algebraic objects by Abū Kāmil, the first use of the positional decimal fractions by al-Uqlidisi, the development of the Arabic numerals and an early algebraic symbolism in the Maghreb, the Thabit number and Thābit theorem by Thābit ibn Qurra, the discovery of several new trigonometric identities by Ibn Yunus and al-Battani, the mathematical proof for Ceva's theorem by Ibn Hűd, the first accurate lunar model by Ibn al- Shatir, the invention of the torquetum by Jabir ibn Aflah, the invention of the universal astrolabe and the equatorium by al-Zarqali, the first description of the crankshaft by al-Jazari, the anticipation of the inertia concept by Averroes, the discovery of the physical reaction by Avempace, the identification of more than 200 new plants by Ibn al-Baitar the Arab Agricultural Revolution, and the Tabula Rogeriana, which was the most accurate world map in pre-modern times by al-Idrisi.Bacharach, 2006, p. 140.
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A discovery made in 1672-1673 by Jean Richer turned the attention of mathematicians to the deviation of the Earth's shape from a spherical form. This astronomer, having been sent by the Academy of Sciences of Paris to Cayenne, in South America, for the purpose of investigating the amount of astronomical refraction and other astronomical objects, notably the parallax of Mars between Paris and Cayenne in order to determine the Earth-Sun distance, observed that his clock, which had been regulated at Paris to beat seconds, lost about two minutes and a half daily at Cayenne, and that in order to bring it to measure mean solar time it was necessary to shorten the pendulum by more than a line (about 1⁄12th of an in.). This fact was scarcely credited till it had been confirmed by the subsequent observations of Varin and Deshayes on the coasts of Africa and America. In South America Bouguer noticed, as did George Everest in the 19th century Great Trigonometric Survey of India, that the astronomical vertical tended to be pulled in the direction of large mountain ranges, due to the gravitational attraction of these huge piles of rock.
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He also discussed human geography and the planetary habitability of the Earth. He also calculated the latitude of Kath, Khwarezm, using the maximum altitude of the Sun, and solved a complex geodesic equation in order to accurately compute the Earth's circumference, which was close to modern values of the Earth's circumference. His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations, based on the angle between a plain and mountain top, which yielded more accurate measurements of the Earth's circumference, and made it possible for it to be measured by a single person from a single location. Self portrait of Alexander von Humboldt, one of the early pioneers of geography as an academic subject in modern sense The European Age of Discovery during the 16th and the 17th centuries, where many new lands were discovered and accounts by European explorers such as Christopher Columbus, Marco Polo, and James Cook revived a desire for both accurate geographic detail, and more solid theoretical foundations in Europe.
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On April 13, they released their 3rd EP, I.F. On April 20, they held auditions for BEJ48, GNZ48 and SHY48 third-generation members. On April 29, they announced 14 second-generation members during GNZ48's first anniversary. On July 28, GNZ48 released their first documentary, Documentary of GNZ48: 7/48 (GNZ48 四十八分之七). On July 29, GNZ48, along with SNH48, BEJ48, SHY48 and CKG48, held auditions for SNH48 ninth-generation, BEJ48, GNZ48 and SHY48 fourth-generation and CKG48 second-generation members. On August 11, Team G began its third stage, "Two-Faced Idol"(Chinese:双面偶像). On September 15, Team Z premiered the performance of "Code, The west of LinHe"(Chinese:代号·林和西). On September 16, Team Z announced Nong Yanping as Captain and Long Yirui as Vice-Captain; they also announced that the team's stage “Trigonometric function”(Chinese:三角函数) will be held at the end of the year. On September 23, they released their 4th EP, Say No. The title track's lyrics were written by members Feng Jiaxi and Luo Hanyue, and its music video was done as a collaboration with the Guangdong Southern Tigers. On October 1, GNZ48 Phase 3 students premiered the "Idol Research Program"(Chinese:偶像研究计划) performance.
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