Math Trig Functions

1. Mathematics Trig Functions

1. Sin and Cos Transformations. Here’s a general formula in order to transform a sin or cos function, as well as the remaining four trig functions.Note that sometimes you’ll see the formula arranged differently; for example, with “ (a )” being the vertical shift at the beginning.
2. Those graphs. Learn how to construct trigonometric functions from their graphs or other features. Trig word problem: modeling annual temperature.

Basis of trigonometry: if two have equal, they are, so their side lengths. Proportionality are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles. In, the trigonometric functions (also called circular functions, angle functions or goniometric functions ) are of an.

They relate the angles of a to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the,. In the context of the standard (a circle with 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the x-component (the adjacent of the angle or the run), and the tangent function gives the slope ( y-component divided by the x-component). For angles less than a, trigonometric functions are commonly defined as of two sides of a right triangle containing the angle, and their values can be found in the lengths of various line segments around a unit circle.

How can the answer be improved?

Modern definitions express trigonometric functions as or as solutions of certain, allowing the extension of the arguments to the whole number line and to the. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a into coordinates. The sine and cosine functions are also commonly used to model phenomena such as and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.

In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations. Main article: Prior to computers, people typically evaluated trigonometric functions by from a detailed table of their values, calculated to many. Such tables have been available for as long as trigonometric functions have been described (see below), and were typically generated by repeated application of the half-angle and angle-addition starting from a known value (such as sin( π / 2) = 1). Modern computers use a variety of techniques.

Mathematics Trig Functions

One common method, especially on higher-end processors with units, is to combine a or (such as, best uniform approximation, and, and typically for higher or variable precisions, and ) with range reduction and a —they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Devices that lack often use an algorithm called (as well as related techniques), which uses only addition, subtraction,. These methods are commonly implemented in for performance reasons. For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the, which itself approximates the trigonometric function by the. Main article: While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period.

The function was discovered by of (180–125 BCE) and of (90–165 CE). The functions sine and cosine can be traced to the functions used in (, ), via translation from Sanskrit to Arabic and then from Arabic to Latin. All six trigonometric functions in current use were known in by the 9th century, as was the law of sines, used in. Produced tables of sines, cosines and tangents.

They were studied by authors including, (14th century), (14th century), (1464), and Rheticus' student. 1400) made early strides in the of trigonometric functions in terms of. The terms tangent and secant were first introduced by the Danish mathematician in his book Geometria rotundi (1583). The first published use of the abbreviations sin, cos, and tan is probably by the French mathematician.

In a paper published in 1682, proved that sin x is not an of x. 's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting ', as well as near-modern abbreviations ( sin., cos., tang., cot., sec., and cosec.). A few functions were common historically, but are now seldom used, such as the ( crd( θ) = 2 sin( θ / 2)), the ( versin( θ) = 1 − cos( θ) = 2 sin 2( θ / 2)) (which appeared in the earliest tables ), the ( coversin( θ) = 1 − sin( θ) = versin( π / 2- θ)), the ( haversin( θ) = 1 / 2versin( θ) = sin 2( θ / 2)), the ( exsec( θ) = sec( θ) − 1), and the ( excsc( θ) = exsec( π / 2 − θ) = csc( θ) − 1). See for more relations between these functions. Etymology The word sine derives from, meaning 'bend; bay', and more specifically 'the hanging fold of the upper part of a ', 'the bosom of a garment', which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning 'pocket' or 'fold' in the twelfth-century translations of works by and into. The choice was based on a misreading of the Arabic written form j-y-b ( ), which itself originated as a from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to 'bowstring', being in turn adopted from 'string'. The word tangent comes from Latin tangens meaning 'touching', since the line touches the circle of unit radius, whereas secant stems from Latin secans—'cutting'—since the line cuts the circle.

The prefix ' (in 'cosine', 'cotangent', 'cosecant') is found in 's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the ) and proceeds to define the cotangens similarly. See also. (1983) June 1964. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.).

Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications., Complex Analysis: an introduction to the theory of analytic functions of one complex variable, second edition, New York, 1966., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991). Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed., London. Kantabutra, Vitit, 'On hardware for computing exponential and trigonometric functions,' IEEE Trans. Computers 45 (3), 328–339 (1996). Maor, Eli, Princeton Univ.

Reprint edition (February 25, 2002):. Needham, Tristan, ' to.

Oxford University Press, (1999). Nielsen, Kaj L.

(1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York, USA:,. O'Connor, J.

O'Connor, J. Pearce, Ian G.,. Weisstein, Eric W., from, accessed 21 January 2006.

External links Wikibooks has a book on the topic of:., ed. (2001) 1994, Springer Science+Business Media B.V. / Kluwer Academic Publishers,. Visualization of the unit circle, trigonometric and hyperbolic functions.

Article about the of sin at. Article about the of cos at.