Shape of Sine Function

Theorem
The sine function is:


 * continuous on the whole of $$\R$$;


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


 * strictly 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 $$\left[{\pi \, . \, . \, 2 \pi}\right]$$;


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


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

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 $$\sin \left({x + \frac \pi 2}\right) = \cos x$$.

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