Properties of the Cosine Function

Theorem: Parity of the Real Cosine Function

The real cosine function is :

$\cos (-x)=\cos (x)$

PROOF

TODO

Theorem: Image of the Real Cosine Function

The image of the real cosine function is $[-1;1]$.

$\cos (\mathbb{R})=[-1;1]$

PROOF

TODO

Theorem: Periodicity of the Real Cosine Function

The real cosine function has a period of $2\pi$. More generally,

$\cos (x+2k\pi )=\cos (x)\forall k\in \mathbb{Z}$

PROOF

TODO

Theorem: Antiperiodicity of the Real Cosine Function

The real cosine function has an antiperiod of $\pi$. More generally,

$\cos (x+(2k+1)\pi )=-\cos (x)k\in \mathbb{Z}$

PROOF

TODO