**Cosine** is a trigonometric ratio. In a right triangle with an angle ,

is the side of the triangle that is part of angle , and is the side opposite the right angle.

## Properties

The cosine of an angle is the x-coordinate of the point of intersection of said angle and a unit circle.

As a result of Euler's formula, the cosine function can also be represented as

The reciprocal of cosine is secant (abbreviated as ), while its inverse is or . Note that sine is not being raised to the power of -1; this is an inverse function, not a reciprocal.

The derivative of is , while its antiderivative is .

The trigonometric addition/subtraction identiy of cosine is:

The square of cosine:

Imaginary number:

### Limits

## Power series

More formally defined, the cosine is defined by the power series, which is convergent for all ,

Where denotes the factorial of n. From this, the above properties may be derived.