y=∣sin⁡x∣y = |\sin x|

Graph of the Function y=∣sin⁡x∣y = |\sin x|

y=∣sin⁡x∣y = |\sin x| is the absolute value of the sine sin⁡x\sin x. The negative half-waves are folded up onto the positive side, giving a graph of arches in a row.

Definition

∣sin⁡x∣={sin⁡x(sin⁡x≥0)−sin⁡x(sin⁡x<0)|\sin x| = \begin{cases} \sin x & (\sin x \geq 0) \\ -\sin x & (\sin x < 0) \end{cases}

Domain and range

  • The domain is all real numbers
  • The range is 0≤y≤10 \leq y \leq 1
  • The period is π\pi
  • It is an even function

Periodicity

The sine has period 2π2\pi, but the absolute value folds the negative half-wave onto the positive side, halving the period to π\pi. Indeed ∣sin⁡(x+π)∣=∣−sin⁡x∣=∣sin⁡x∣|\sin(x + \pi)| = |-\sin x| = |\sin x|.

Symmetry

Since ∣sin⁡(−x)∣=∣−sin⁡x∣=∣sin⁡x∣|\sin(-x)| = |-\sin x| = |\sin x|, the function is even and symmetric about the yy-axis. It is symmetric about each line x=π2+nπx = \dfrac{\pi}{2} + n\pi through the top of an arch as well.

The corners

On each interval the curve rises from 00 to a maximum of 11 and falls back to 00, repeating the same arch. The maximum 11 is attained at x=π2+nπx = \dfrac{\pi}{2} + n\pi.

At x=nπx = n\pi the value is 00, but the slope reverses sign there, producing a sharp corner where the function is not differentiable. Unlike the smooth sine wave, the points touching the xx-axis are corners.

Approach to x=0x = 0Slope
from the right, x→0+x \to 0^{+}11
from the left, x→0−x \to 0^{-}−1-1

The function is continuous everywhere but fails to be differentiable at the corners x=nπx = n\pi. It is a familiar example of a function that is continuous without being differentiable.

Notable values

xx∣sin⁡x∣|\sin x|
0000
π6\dfrac{\pi}{6}12\dfrac{1}{2}
π2\dfrac{\pi}{2}11
5π6\dfrac{5\pi}{6}12\dfrac{1}{2}
π\pi00

Average value and Fourier series

The average height over one period is as follows.

1π∫0πsin⁡x dx=2π≈0.6366\frac{1}{\pi}\int_0^{\pi}\sin x\,dx = \frac{2}{\pi} \approx 0.6366

Expanding in a Fourier series makes that average appear as the constant term.

∣sin⁡x∣=2π−4π∑k=1∞cos⁡2kx4k2−1|\sin x| = \frac{2}{\pi} - \frac{4}{\pi}\sum_{k=1}^{\infty}\frac{\cos 2kx}{4k^2 - 1}

Being even, the function has no sine terms, and since the period is π\pi, only the even multiples of the frequency appear.

Applications

The waveform of full-wave rectification, which converts alternating current toward direct current, is exactly the shape of ∣sin⁡x∣|\sin x|1. The average value 2π\dfrac{2}{\pi} gives the rough output voltage after rectification.

  • The output waveform of a rectifier circuit
  • A model for a periodic signal with kinks
  • An exercise in differentiating and integrating absolute values combined with trigonometric functions
  1. Rectifier, Wikipedia