When Undoing Does Not Undo

$\sin\left(\arcsin\left(c\right)\right) = c$ for every c from −1 to 1: arcsin finds an angle with that sine, and sine reads it back. The other order fails. $\arcsin\left(\sin\left(150^{\circ}\right)\right)$ is not 150°: $\sin\left(150^{\circ}\right) = \frac{1}{2}$, and arcsin(1/2) is 30°, the angle in its range.

So arcsin(sin θ) = θ only for θ in [−90°, 90°], arccos(cos θ) = θ only for θ in [0°, 180°], and arctan(tan θ) = θ only for θ in (−90°, 90°). Outside those, work the inside out first, then apply the inverse.