Shape of Sine Function

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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 and Cosine are 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.


Graph of Sine Function

Sine.png


$\blacksquare$


Also see


Sources