Trigonometric function

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.

|**Function**|**Abbreviation**|**Relation (Radians)**|
| --- | --- | --- |
|**Sine**|sin||
|**Cosine**|cos||
|**Tangent**|tan (or tg)||
|**Cotangent**|cot (or ctg)||
|**Secant**|sec||
|**Cosecant**|csc (or cosec)||

## 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

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
Note that since all those triangles are similar, this ratio does not depend on the particular right triangle that is chosen, as long as it contains the angle _A_.

The set of zeroes of sine (that is, the values of  for which ) is

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

The set of zeroes of cosine is

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

The set of zeroes of tangent is
This is the same set as that of the sine function, since

The remaining three functions are best defined using the above three functions.

4) The **cosecant** csc(_A_) is the multiplicative inverse of sin(_A_); it is the ratio of the length of the hypotenuse to the length of the opposite side:
- .

5) The **secant** sec(_A_) is the multiplicative inverse of cos(_A_); it is the ratio of the length of the hypotenuse to the length of the adjacent side:
- .

6) The **cotangent** cot(_A_) is the multiplicative inverse of tan(_A_); it is the ratio of the length of the adjacent side to the length of the opposite side:
- .

### Definitions by power series

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

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

## Identities

Some important identities:

## 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:
- Hyperbolic cosine:
- Hyperbolic tangent:
- Hyperbolic cotangent:
- Hyperbolic secant:
- Hyperbolic cosecant:

## Related pages

- Periodic function
- Pythagorean theorem
- Sine wave
- Trigonometry

## 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