Zeroes of Sine and Cosine/Cosine

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Theorem

Let $x \in \R$.


$\cos x = 0$ if and only if $x = \paren {n + \dfrac 1 2} \pi$ for some $n \in \Z$.


Proof

From Sine and Cosine are Periodic on Reals: Corollary:

$\cos x$ is:

strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$

and:

strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$


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

$\blacksquare$