## FANDOM

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Cosine is a trigonometric ratio. In a right triangle with an angle $\theta$ ,

$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$

$\text{Adjacent}$ is the side of the triangle that is part of angle $\theta$ , and $\text{hypotenuse}$ 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

$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}$

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

The derivative of $\cos(x)$ is $-\sin(x)$ , while its antiderivative is $\sin(x)$ .

The trigonometric addition/subtraction identiy of cosine is:

$\cos(a\pm b)=\cos(a)\cos(b)\mp\sin(a)\sin(b)$

The square of cosine:

$\cos^2(\theta)=\frac{\cos(2\theta)+1}{2}$

Imaginary number:

$\cos(ix)=\cosh(x)$

## Power series

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

$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$

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