# Is the function f(x) = cos x even, odd or neither?

Oct 31, 2015

Even

#### Explanation:

$\cos \left(x\right) = \cos \left(- x\right)$, therefore cosine is an even function.

http://mathworld.wolfram.com/Cosine.html

Jan 8, 2018

To prove that $\cos \left(\theta\right)$ is even, i.e. that $\cos \left(- \theta\right) = \cos \left(\theta\right)$, we can use the unit circle, which mind you, is the definition of cosine arguments outside the interval $\left[0 , \frac{\pi}{2}\right]$.

The unit circle is a circle of radius one centered at the origin. We can draw the following constructions for $\cos \left(\theta\right)$ and $\cos \left(- \theta\right)$:

We see that the points $\left(\cos \left(\theta\right) , \sin \left(\theta\right)\right)$ and $\left(\cos \left(- \theta\right) , \sin \left(- \theta\right)\right)$ are on the same vertical line. Since the unit circle is in a cartesian coordinate system, this must mean they have the same $x$-coordinates.

The first point has an $x$-coordinate of $\cos \left(\theta\right)$, and the second has an $x$-coordinate of $\cos \left(- \theta\right)$, and they must be equal, so it quite easily follows that:
$\cos \left(- \theta\right) = \cos \left(\theta\right)$

Which proves that cosine is an even function.