# Cosine

One of the trigonometric functions:

$$y=\cos x.$$

Its domain of definition is the entire real line; its range of values is the closed interval $[-1,1]$; the cosine is an even periodic function (with period $2\pi$). The cosine and the sine are related via the formula

$$\sin^2x+\cos^2x=1.$$

The cosine and the secant are related via the formula

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

The derivative of the cosine is:

$$(\cos x)'=-\sin x.$$

The integral of the cosine is:

$$\int\cos(x)\,dx=\sin x+C.$$

The series expansion is:

$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dotsb,\qquad-\infty<x<\infty.$$

The inverse function is the arccosine.

The cosine and sine of a complex argument $z$ are related to the exponential function by Euler's formula:

$$e^{iz}=\cos z+i\sin z.$$

If $x$ is a real number, then

$$\cos x=\frac{e^{ix}+e^{-ix}}{2}.$$

If $z=ix$ (a purely imaginary number), then

$$\cos ix=\frac{e^x+e^{-x}}{2}=\cosh x,$$

where $\cosh x$ is the hyperbolic cosine.

A geometric interpretation of the cosine of an argument (angle) $\phi$ is as follows. Consider the unit circle $T$ in the (complex) plane with origin $0$. Let $\phi$ denote the angle between the radius (thought of as varying) and the positive $x$-axis. Then $\cos\phi$ is equal to the (signed) distance from the point $e^{i\phi}$ on $T$ corresponding to $\phi$ to the $x$-axis. See also Sine.