When Undoing Does Not Undo

sin(arcsin(c)) = 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(sin(150°)) is not 150°: sin(150°) = 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.