Graph of the Function $y = |\sin x|$

The function $y = |\sin x|$ is the absolute value of the sine. Its negative half-waves are folded up to the positive side, giving a graph of repeated arches.

Definition

Domain and range

Its domain is all real numbers. Being an absolute value, it is always non-negative, so its range is $0 \le y \le 1$.

Periodicity

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

Symmetry

Since $|\sin(-x)| = |-\sin x| = |\sin x|$, it is an even function, symmetric about the $y$-axis.

Behavior and corners

On each interval it rises from $0$ to a maximum of $1$ and back to $0$, forming a repeated arch. The maximum $1$ occurs at $x = \dfrac{\pi}{2} + n\pi$. At $x = n\pi$ the value is $0$, but there the slope reverses sign, producing a sharp corner where the function is not differentiable. Unlike the smooth sine wave, the points touching the $x$-axis are corners.

Relation to other functions

The waveform of full-wave rectification, which converts alternating current toward direct current, is exactly the shape of $|\sin x|$. Its Fourier series is

a sum of even-frequency cosines about the average value $\dfrac{2}{\pi}$.

Specific values and continuity

For example $\left|\sin\dfrac{\pi}{6}\right| = \dfrac{1}{2}$ and $\left|\sin\dfrac{\pi}{2}\right| = 1$. The function is continuous everywhere but fails to be differentiable only at the corners $x = n\pi$. Its average height over one period is $\dfrac{2}{\pi} \approx 0.64$, matching the constant term of the Fourier series above.

Applications

Besides the analysis of rectifier circuits, it serves as a model for periodic signals with kinks and as a classic exercise in differentiating and integrating combinations of absolute values and trigonometric functions.