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# Cosine

The cosine of an angle ${\displaystyle \scriptstyle \theta \,}$, denoted as ${\displaystyle \scriptstyle \cos \theta \,}$, is one of the most important [circular] trigonometric functions.

## Contents

Given the angle ${\displaystyle \scriptstyle \theta \,}$ of an arc on a unit circle, ${\displaystyle \scriptstyle \cos \theta \,}$ is the length of the side on a right triangle going along a line along ${\displaystyle x=0}$ (the other two sides being a side opposing a vertex coinciding with the center of the circle and hypotenuse).

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Per the Pythagorean theorem, ${\displaystyle \scriptstyle (\sin \theta )^{2}+(\cos \theta )^{2}\,=\,1\,}$.

The following graph shows the sine and cosine functions graphed together (sine is in red, cosine in blue):

## Taylor series expansion

The Taylor series expansion of the cosine function is

${\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots .\,}$

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