Sine-squared function

Definition

This function, denoted $\sin ^{2}$, is defined as the composite of the square function and the sine function. Explicitly, it is the map:

$x\mapsto (\sin x)^{2}$

For brevity, we write $(\sin x)^{2}$ as $\sin ^{2}x$.

Key data

Item	Value
Default domain	all real numbers, i.e., all of $\mathbb {R}$
range	$[0,1]$, i.e., $\{y\mid 0\leq y\leq 1\}$ absolute maximum value: 1, absolute minimum value: 0
period	$\pi$, i.e., $180\,^{\circ }$
local maximum value and points of attainment	All local maximum values are equal to 1, and are attained at odd integer multiples of $\pi /2$.
local minimum value and points of attainment	All local minimum values are equal to 0, and are attained at integer multiples of $\pi$.
points of inflection (both coordinates)	odd multiples of $\pi /4$, with value 1/2 at each point.
derivative	$x\mapsto \sin(2x)=2\sin x\cos x$, i.e., double-angle sine function.
second derivative	$x\mapsto -2\cos(2x)$
$n^{th}$ derivative	$2^{n-1}$ times an expression that is $\pm \sin$ or $\pm \cos$ of $2x$, depending on the remainder of $n$ mod $4$
antiderivative	$x\mapsto {\frac {x}{2}}-{\frac {\sin(2x)}{4}}+C$
mean value over a period	1/2
expression as a sinusoidal function plus a constant function	$(1/2)-\cos(2x)/2$
interval description based on increase/decrease and concave up/down	For each integer $n$, the interval from $n\pi$ to $(n+1)\pi$ is subdivided into four pieces: $(n\pi ,n\pi +\pi /4)$: increasing and concave up $(n\pi +\pi /4,n\pi +\pi /2)$: increasing and concave down $(n\pi +\pi /2,n\pi +3\pi /4)$: decreasing and concave down, $(n\pi +3\pi /4,(n+1)\pi )$: decreasing and concave up