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

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.

$\blacksquare$

Also see

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