# Why is cos(pi) and cos (-pi) both equal to -1?

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Alan P. Share
Apr 22, 2015

A full circle is $2 \pi$

$\pi$ is half way around the circle counter-clockwise.

$- \pi$ is half way around the circle clockwise.

$\cos \left(\pi\right) = \cos \left(- \pi\right)$ are the both $\cos$ values for the same place

They are both equal to $\left(- 1\right)$
because
if viewed as a unit circle centered on the Cartesian origin
the $\cos$ is the $x$ value
and, at halfway around the unit circle,
$x = - 1$

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Mar 30, 2017

This is what a cosine graph looks like. Since $\pi$ in degrees is 180, it is exactly the same distance from 0 degrees.

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graph{cos(x) [-4.76, 4.84, -2.64, 2.16]}

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Antoine Share
Apr 22, 2015

One of the reasons is because $\cos$ is an even function.

This means that $\cos \theta = \cos \left(- \theta\right)$

An example is the present case $\cos \pi = \cos \left(- \pi\right) = - 1$

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