Shape of Sine Function

Theorem
The sine function is:


 * $(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$


 * $(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$


 * $(3): \quad$ concave on the interval $\closedint 0 \pi$


 * $(4): \quad$ convex on the interval $\closedint \pi {2 \pi}$

Proof
From the discussion of Sine is Periodic on Reals, we have that:
 * $\sin \paren {x + \dfrac \pi 2} = \cos x$

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

Also see

 * Properties of Real Sine Function


 * Shape of Cosine Function
 * Shape of Tangent Function
 * Shape of Cotangent Function
 * Shape of Secant Function
 * Shape of Cosecant Function