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One of the [[Trigonometric functions|trigonometric functions]]:
 
One of the [[Trigonometric functions|trigonometric functions]]:
  
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$$y=\cos x.$$
  
Its domain of definition is the entire real line; its range of values is the closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c0266302.png" />; the cosine is an even periodic function (with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c0266303.png" />). The cosine and the sine are related via the formula
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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
  
<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/c026630/c0266304.png" /></td> </tr></table>
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$$\sin^2x+\cos^2x=1.$$
  
 
The cosine and the secant are related via the formula
 
The cosine and the secant are related via the formula
  
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$$\cos x=\frac{1}{\sec x}.$$
  
 
The derivative of the cosine is:
 
The derivative of the cosine is:
  
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$$(\cos x)'=-\sin x.$$
  
 
The integral of the cosine is:
 
The integral of the cosine is:
  
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$$\int\cos xdx=\sin x+C.$$
  
 
The series expansion is:
 
The series expansion is:
  
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$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots,\quad-\infty<x<\infty.$$
  
 
The inverse function is the arccosine.
 
The inverse function is the arccosine.
  
The cosine and sine of a complex argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c0266309.png" /> are related to the exponential function by Euler's formula:
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The cosine and sine of a complex argument $z$ are related to the exponential function by Euler's formula:
  
<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/c026630/c02663010.png" /></td> </tr></table>
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$$e^{iz}=\cos z+i\sin z.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663011.png" /> is a real number, then
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If $x$ is a real number, then
  
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$$\cos x=\frac{e^{ix}+e^{-ix}}{2}.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663013.png" /> (a purely imaginary number), then
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If $z=ix$ (a purely imaginary number), then
  
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$$\cos ix=\frac{e^x+e^{-x}}{2}=\cosh x,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663015.png" /> is the hyperbolic cosine.
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where $\cosh x$ is the hyperbolic cosine.
  
  
  
 
====Comments====
 
====Comments====
A geometric interpretation of the cosine of an argument (angle) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663016.png" /> is as follows. Consider the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663017.png" /> in the (complex) plane with origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663019.png" /> denote the angle between the radius (thought of as varying) and the positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663020.png" />-axis. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663021.png" /> is equal to the (signed) distance from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663023.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663024.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026630/c02663025.png" />-axis. See also [[Sine|Sine]].
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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|Sine]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>

Revision as of 14:01, 23 April 2014

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:

$$\int\cos xdx=\sin x+C.$$

The series expansion is:

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

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.


Comments

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.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Cosine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cosine&oldid=31900
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