# Difference between revisions of "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 $\sin^2x$.

## Key data

Item Value
Default domain all real numbers, i.e., all of $\R$
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 $(1/2) - \cos(2x)/2$
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