Zeroes of Sine and Cosine/Sine

Theorem
Let $x \in \R$.


 * $\sin x = 0$, $x = n \pi$ for some $n \in \Z$.

Proof
From Sine and Cosine are Periodic on Reals: Corollary:

$\sin x$ is:
 * strictly positive on the interval $\openint 0 \pi$

and:
 * strictly negative on the interval $\openint \pi {2 \pi}$

The result follows directly from Sine and Cosine are Periodic on Reals.