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One of the trigonometric functions:

Its domain of definition is the entire real line; its range of values is the closed interval ; the cosine is an even periodic function (with period ). The cosine and the sine are related via the formula

The cosine and the secant are related via the formula

The derivative of the cosine is:

The integral of the cosine is:

The series expansion is:

The inverse function is the arccosine.

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

If is a real number, then

If (a purely imaginary number), then

where is the hyperbolic cosine.


A geometric interpretation of the cosine of an argument (angle) is as follows. Consider the unit circle in the (complex) plane with origin . Let denote the angle between the radius (thought of as varying) and the positive -axis. Then is equal to the (signed) distance from the point on corresponding to to the -axis. See also Sine.


[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:
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