Shape of Sine Function

Theorem
The sine function is:


 * continuous on the whole of $\R$


 * strictly increasing on the interval $\left[{-\dfrac \pi 2 \, . \, . \, \dfrac \pi 2}\right]$


 * strictly decreasing on the interval $\left[{\dfrac \pi 2 \, . \, . \, \dfrac {3 \pi} 2}\right]$


 * concave on the interval $\left[{0 \, . \, . \, \pi}\right]$


 * convex on the interval $\left[{\pi \, . \, . \, 2 \pi}\right]$


 * $\forall n \in \Z: \sin \left({2 n + \dfrac 1 2}\right) \pi = 1$


 * $\forall n \in \Z: \sin \left({2 n - \dfrac 1 2}\right) \pi = -1$

Proof
The fact of the continuity of $\sin x$ is established in the discussion of Derivative of Sine Function.

From the discussion of Sine and Cosine are Periodic on Reals, we have that $\sin \left({x + \dfrac \pi 2}\right) = \cos x$.

The result then follows directly from the Nature of Cosine Function.

Also see

 * Nature of Cosine Function