Trigonometric function

can be constructed using a circle centered at *O* with  radius of 1.]]

In mathematics, the **trigonometric functions** are a set of functions which relate angles to the sides of a right triangle. There are many trigonometric functions, the 3 most common being sine, cosine, tangent, followed by cotangent, secant and cosecant. The last three are called **reciprocal trigonometric functions,** because they act as the reciprocals of other functions. Secant and cosecant are rarely used.

## Definition 
The **trigonometric functions** sometimes are also called circular functions. They are functions of an angle; they are important when studying triangles, 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 (a circle with radius of one).

### Right triangle definitions 
 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 a 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, also 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 right angle the angle we are interested in, hence its name. In this case, the adjacent side is **b**.

All triangles are taken to exist in Euclidean geometry, 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. Notice that strictly speaking, the following definitions 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

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

## 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
:\sin 2x = 2 \sin x \cos x
:\cos 2x = \cos x \cos x - \sin x \sin x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^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}

## Hyperbolic functions
The **hyperbolic functions** are like the trigonometric functions, in that they have very similar properties. Each of six trigonometric functions has a corresponding hyperbolic form. They are defined in terms of the exponential function, which is based on the constant *e*.
* Hyperbolic sine:
::\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.
* Hyperbolic cosine:
::\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.
* Hyperbolic tangent:
::\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}.
* Hyperbolic cotangent:
::\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} =  \frac{e^{2x} + 1} {e^{2x} - 1} = \frac{1 + e^{-2x}} {1 - e^{-2x}}, \qquad x \neq 0.
* Hyperbolic secant:
::\operatorname{sech}\,x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = \frac{2e^x} {e^{2x} + 1} = \frac{2e^{-x}} {1 + e^{-2x}}.
* Hyperbolic cosecant: 
::\operatorname{csch}\,x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = \frac{2e^x} {e^{2x} - 1} = \frac{2e^{-x}} {1 - e^{-2x}}, \qquad x \neq 0.

## Related pages
* Periodic function
* Pythagorean theorem
* Sine wave
* Trigonometry

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

## 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.
*An excellent Flash animation for learning the unit circle