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Difference between revisions of "Cotangent"

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One of the [[Trigonometric functions|trigonometric functions]]:
 
One of the [[Trigonometric functions|trigonometric functions]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266901.png" /></td> </tr></table>
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$$y=\operatorname{cotan}x=\frac{\cos x}{\sin x};$$
  
other notations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266904.png" />. The domain of definition is the entire real line with the exception of the points with abscissas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266906.png" />. The cotangent is an unbounded odd periodic function (with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266907.png" />). The cotangent and the tangent are related by
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266908.png" /></td> </tr></table>
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$$\operatorname{cotan}x=\frac{1}{\tan x}.$$
  
 
The inverse function to the cotangent is called the arccotangent. The derivative of the cotangent is given by:
 
The inverse function to the cotangent is called the arccotangent. The derivative of the cotangent is given by:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c0266909.png" /></td> </tr></table>
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$$(\operatorname{cotan}x)'=\frac{-1}{\sin^2x}.$$
  
 
The integral of the cotangent is given by:
 
The integral of the cotangent is given by:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c02669010.png" /></td> </tr></table>
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$$\int\operatorname{cotan}xdx=\ln|{\sin x}|+C.$$
  
 
The series expansion is:
 
The series expansion is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c02669011.png" /></td> </tr></table>
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$$\operatorname{cotan}x=\frac1x-\frac x3-\frac{x^3}{45}-\ldots,\quad0<|x|<\pi.$$
  
The cotangent of a complex argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c02669012.png" /> is a meromorphic function with poles at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c02669013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026690/c02669014.png" />.
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The cotangent of a complex argument $z$ is a meromorphic function with poles at the points $z=\pi n$, $n=0,\pm1,\pm2,\ldots$.
  
  

Revision as of 13:49, 23 April 2014

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:

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

The series expansion is:

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

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=31899
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article