Sine-squared function

This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki

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., $\{y\mid 0\le y\le 1\}$ 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	$(1/2)-\cos (2x)/2$
important symmetries	even function more generally, miror symmetry about any vertical line of the form $x=n\pi /2$, $n$ an integer. Also, half turn symmetry about all points of the form $(n\pi /2+\pi /4,1/2)$.
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