Trigonometric functionIn mathematics, the trigonometric functions are functionss of an angle, important when studying triangless and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of coordinates of points on the unit circle, or, more generally still, as infinite series, or equally generally, as solutions of certain differential equations. All four approaches will be presented below. These are the six basic trigonometric functions, together with their standard notational abbreviations. The last four functions are defined in terms of the first two. In other words, the four equations below are definitions, not proved identities.
See Also\n* Exact_trigonometric_constantsUnit-circle definitionsThe six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles we have used so far. The equation for the unit circle is:
 \nAlternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at right), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see below). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.)
Here is a plot of sine and cosine:
 Series definitions\n
Please note: Here, and generally in calculus, it is of utmost importance that all angles are measured in radians.
Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is negative sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x:
Relationship to exponential functionBy allowing the arguments of the sine and cosine functions to be complex, it can be shown that they are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:
Definitions via differential equationsBoth the sine and cosine functions satisfy the differential equationInverse functionsThe trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principle inverses are usually defined as:
Identities
Properties and applicationsThe trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results: The law of sines for an arbitrary triangle states:\n:\nIt can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number sin(A)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points A, B and C. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance. If the angle is not contained between the two sides, the triangle may not be unique. Be aware of this ambiguous case of the Sine Law. The law of cosines is an extension of the Pythagorean theorem:
The image on the right displays a two-dimensional graph based on such a summation of sines and cosines, illustrating the fact that arbitrarily complicated closed curvess can be described by a Fourier series. Its equation is:\n: \nwhere F(n) is the nth Fibonacci number.
For a compilation of many relations between the trigonometric functions, see trigonometric identities.
HistoryThe earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A'\' + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula sin(A/2)2 = (1 − cos(A''))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day. The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th–5th century), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees. The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values. Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhatiya) was transliterated by the Arabs as jiba, but was confused for another word, jaib ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, perhaps because jiba had been abbreviated jb in Arabic. All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by the De triangulis omnimodus (1464) of Regiomontanus (1436–1476), as well as his later Tabulae directionum (which included the tangent function, unnamed). The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was the first 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. The Introductio in analysin infinitorum (1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, defining them as infinite series and presenting "Euler's formula" eix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec..References\n* Carl B. Boyer, A History of Mathematics, 2nd ed. (Wiley, New York, 1991).\n* Eli Maor, Trigonometric Delights (Princeton Univ. Press, 1998).\n* "Trigonometric functions", MacTutor History of Mathematics Archive.\n* Tristan Needham, Visual Complex Analysis, (Oxford University Press, 2000), ISBN 0198534469 Book websiteSee also\n* Generating trigonometric tables\n* Hyperbolic function\n* Pythagoras\n* Pythagorean theorem\n* Trigonometric identity\n* What is trigonometry used forExternal link\n*Dave's Short Trig Course uses interactive java applets that are very helpful for learning about trig functions on the unit circle, as well as being useful more generally for students of trigonometry. Category:Trigonometry Category:Wikipedia Featured Articles\n\n\n\n\n\n |
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In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A:
We use the following names for the sides of the triangle:\n* The hypotenuse is the side opposite the right angle, in this case h.\n* The opposite side is the side opposite to the angle we are interested in, in this case a.\n* The adjacent side is the side that is a leg of the angle, but not the hypotenuse, in this case b.
Then,
1). The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case\n:sin(A) = opp/hyp = a/h.\nNote that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.
2). The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case\n:cos(A) = adj/hyp = b/h.
3). The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case\n:tan(A) = opp/adj = a/b.
The remaining three functions are best defined using the above three functions.
4). The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:\n:csc(A) = hyp/opp = h/a.
5). The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:\n:sec(A) = hyp/adj = h/b.
6). The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:\n:cot(A) = adj/opp = b/a.
