Sine and Cosine are Periodic on Reals/Corollary/Cosine

Corollary to Sine and Cosine are Periodic on Reals

Let $x \in \R$.

$\cos x$ is strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ and strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$

Proof

From the discussion in the proof of Real Cosine Function is Periodic:

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$\map \sin {x + \eta} = \cos x$

$\map \cos {x + \eta} = -\sin x$

for $\eta \in \R_{>0}$, where $\pi$ was defined as $\pi := 2 \eta$.

It follows that $\eta = \dfrac \pi 2$, thus:

$\map \cos {x + \pi} = -\map \sin {x + \dfrac \pi 2} = -\cos x$

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From the discussion in the proof of Real Cosine Function is Periodic:

$\cos \eta = \map \cos {-\eta} = 0$ and $\cos x > 0$ for $-\eta < x < \eta$

It follows directly that:

$\forall x \in \closedint {-\dfrac \pi 2} {\dfrac \pi 2}: \cos x \ge 0$

As $\map \cos {x + \pi} = -\cos x$, we have:

$\forall x \in \closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}: \cos x \le 0$

$\blacksquare$