Sine and Cosine are Periodic on Reals/Sine

Theorem
The sine function is periodic real function:
 * $\exists L \in \R_{\neq 0}: \forall x \in \R: \sin x = \map \sin {x + L}$


 * SineCos.png

Proof
Since cosine is periodic, let $K$ be its period.

Then:
 * $\cos K = \map \cos {0 + K} = \cos 0$

Because Cosine of Zero is One:
 * $\cos K = 1$

Furthermore:

Then, the following holds:

Thus $\sin$ is periodic with some period $L \leq K$.