Sine-squared function

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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