Sine and Cosine are Periodic on Reals/Sine/Proof 1

Theorem
The sine function is periodic with the same period as the cosine function.


 * SineCos.png

Proof
Since Cosine is Periodic on Reals, 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 $L \leq K$.