Shape of Sine Function

Theorem
The sine function is:


 * increasing on the interval $$\left[{-\frac \pi 2 \, . \, . \, \frac \pi 2}\right]$$;
 * decreasing on the interval $$\left[{\frac \pi 2 \, . \, . \, \frac {3\pi} 2}\right]$$;
 * concave on the interval $$\left[{0 \, . \, . \, \pi}\right]$$;
 * convex on the interval $$\left[{\pi \, . \, . \, 2 \pi}\right]$$.

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

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