Sine and Cosine are Periodic on Reals/Corollary/Sine

Corollary to Sine and Cosine are Periodic on Reals
Let $x \in \R$.


 * $\sin x$ is strictly positive on the interval $\openint 0 \pi$ and strictly negative on the interval $\openint \pi {2 \pi}$

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


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

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

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


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

From Sine and Cosine are Periodic on Reals: Corollary: Cosine, it follows that $\cos x$ is strictly positive on $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ and strictly negative on $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$.

It follows directly that:
 * $\forall x \in \closedint 0 \pi: \sin x \ge 0$

and:
 * $\forall x \in \closedint \pi {2 \pi} : \sin x \le 0$