# Cotangent

One of the trigonometric functions:

$$y=\operatorname{cotan}x=\frac{\cos x}{\sin x};$$

other notations are $\cot x$, $\operatorname{cotg}x$ and $\operatorname{ctg}x$. The domain of definition is the entire real line with the exception of the points with abscissas $x=\pi n$, $n=0,\pm1,\pm2,\ldots$. The cotangent is an unbounded odd periodic function (with period $\pi$). The cotangent and the tangent are related by

$$\operatorname{cotan}x=\frac{1}{\tan x}.$$

The inverse function to the cotangent is called the arccotangent. The derivative of the cotangent is given by:

$$(\operatorname{cotan}x)'=\frac{-1}{\sin^2x}.$$

The integral of the cotangent is given by:

$$\int\operatorname{cotan}x\,dx=\ln|{\sin x}|+C.$$

The series expansion is:

$$\operatorname{cotan}x=\frac1x-\frac x3-\frac{x^3}{45}-\dotsb,\quad0<|x|<\pi.$$

The cotangent of a complex argument $z$ is a meromorphic function with poles at the points $z=\pi n$, $n=0,\pm1,\pm2,\ldots$.

#### Comments

See also Tangent, curve of the; Sine; Cosine.

**How to Cite This Entry:**

Cotangent.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cotangent&oldid=44648