Zeroes of Sine and Cosine/Sine

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Theorem

Let $x \in \R$.


$\sin x = 0$, if and only if $x = n \pi$ 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.

$\blacksquare$