Shape of Sine Function

Theorem
The sine function is:


 * $(1): \quad$ strictly increasing on the interval $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$


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


 * $(3): \quad$ concave on the interval $\left[{0 \,.\,.\, \pi}\right]$


 * $(4): \quad$ convex on the interval $\left[{\pi \,.\,.\, 2 \pi}\right]$

Proof

 * Sine.png

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 Shape of Cosine Function.

Also see

 * Basic Properties of Sine Function
 * Shape of Cosine Function
 * Shape of Tangent Function
 * Shape of Cotangent Function