Sine and Cosine are Periodic on Reals/Corollary/Sine
Jump to navigation
Jump to search
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
Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: See similar in Sine and Cosine are Periodic on Reals/Corollary/Cosine You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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:
- $\forall x \in \closedint \pi {2 \pi} : \sin x \le 0$
$\blacksquare$