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.
$\blacksquare$