Shape of Sine Function

From ProofWiki
Jump to: navigation, search

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.


Graph of Sine Function

Sine.png


$\blacksquare$


Also see


Sources