Trigonometric function

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All of the trigonometric functions of any angle can be constructed using a circle centered at O with radius of 1.

Trigonometric functions: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent

In mathematics there are 6 trigonometric functions: sine, cosine, tangent, cotangent, secant and cosecant. Secant and cosecant are rarely used.

Function	Abbreviation	Relation
Sine	sin	$\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) \,$
Cosine	cos	$\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right)\,$
Tangent	tan (or tg)	$\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta} \,$
Cotangent	cot (or ctg)	$\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta} \,$
Secant	sec	$\sec \theta = \frac{1}{\cos \theta} = \csc \left(\frac{\pi}{2} - \theta \right) \,$
Cosecant	csc (or cosec)	$\csc \theta =\frac{1}{\sin \theta} = \sec \left(\frac{\pi}{2} - \theta \right) \,$

1 Definition
- 1.1 Right triangle definitions
- 1.2 Definitions by power series
2 Identities
3 References
4 See also
5 Other websites

[change] Definition

The trigonometric functions sometimes are also called circular functions. They are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle or a circle that has the radius of one.

[change] Right triangle definitions

A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle.

In order to define the trigonometric functions for the angle A, start with an right triangle that contains the angle A:

We use the following names for the sides of the triangle:

The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
The opposite side is the side opposite to the angle we are interested in, in this case a.
The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

All triangles are taken to exist in the Euclidean plane so that the inside angles of each triangle sum to π radians (or 180°); therefore, for a right triangle the two non-right angles are between zero and π/2 radians. The reader should note that the following definitions, strictly speaking, only define the trigonometric functions for angles in this range. We extend them to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.

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

$\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.$

Note 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.

The set of zeroes of sine (i.e., the values of $x$ for which $sin x = 0$ ) is

$\left\{n\pi\big| n\isin\mathbb{Z}\right\}.$

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

$\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}.$

The set of zeroes of cosine is

$\left\{\frac{\pi}{2}+n\pi\bigg| n\isin\mathbb{Z}\right\}.$

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

$\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}.$

The set of zeroes of tangent is

$\left\{n\pi\big| n\isin\mathbb{Z}\right\}.$

The same set of the sine function since

$\tan A = \frac {\sin A}{\cos A}.$

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:

$\csc A = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}$ .

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:

$\sec A = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}$ .

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:

$\cot A = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}$ .

[change] Definitions by power series

One can define the trigonometric functions also by using power series:

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}$

and define tangent, cotangent, secant and cosecant using identities, see below.

[change] Identities

Some important identities:

$\tan x = \frac{\sin x}{\cos x}$

$\cot x = \frac{\cos x}{\sin x}$

$\sec x = \frac{1}{\cos x}$

$\csc x = \frac{1}{\sin x}$

sin 2 x + cos 2 x = 1

sin2 x = 2sin x cos x

cos2 x = cos x cos x - sin x sin x = cos 2 x - sin 2 x = 2cos 2 x - 1 = 1 - 2sin 2 x

$\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$

$\sin \left(x \pm y \right)=\sin x \cos y \pm \cos x \sin y$

$\cos \left(x \pm y \right)=\cos x \cos y \mp \sin x \sin y$

$\tan \left ( x \pm y \right ) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}$

[change] References

Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
"Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2002).
Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.

[change] See also

Wikibooks has more about this subject:

Trigonometry

[change] Other websites

nanoSouffle Online Grapher - Extensible features for graphing functions... works in just about every browser, and JavaScript is simply a plus for real-time updates, and not a requirement.

Sine and cosine function with an implementation in Rexx.
Trigonomic Functions - An interactive sketch showing the trigonometric functions in terms of the unit circle. (Requires Java.)
A chart of trigonometric functions and their values in respect to angles, in both degrees and radians
An excellent Flash animation for learning the unit circle

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Trigonometric function

From the Simple English Wikipedia, the free encyclopedia that anyone can change

Contents

[change] Definition

[change] Right triangle definitions

[change] Definitions by power series

[change] Identities

[change] References

[change] See also

[change] Other websites

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