Inverse Trig and Restricted Domains

Sine hits every value between -1 and 1 infinitely many times. sin(π/6) = 0.5, but so does sin(5pi/6), sin(13pi/6), and endlessly more. A function can only have an inverse if it's one-to-one, so before you can undo sine, you have to restrict it to a piece of its domain where it never repeats a y-value. The standard choice for sine is the interval from -π/2 to π/2: on that stretch sine climbs steadily from -1 to 1, hitting each value exactly once. Restricted to that domain, sine is invertible, and its inverse is arcsin(x), sometimes written sin⁻¹(x).

Because arcsin is the inverse of this restricted sine, its graph is the reflection of that piece of the sine curve over the line y = x. Restricted sine runs from (-π/2, -1) to (π/2, 1); arcsin runs from (-1, -π/2) to (1, π/2), swapping x and y coordinates the way every inverse does. Feed arcsin a ratio between -1 and 1, and it hands back the one angle in [-π/2, π/2] with that sine value. Cosine needs the same treatment but a different window, since cosine isn't one-to-one on [-π/2, π/2] either (cos(-π/4) = cos(π/4)). Cosine is restricted to [0, π] instead, where it steadily falls from 1 to -1 without repeating, and arccos is built from that piece.

This restrict-then-reflect pattern is exactly how all the inverse trig functions get built, including arctan later with its own restricted interval. It also explains why arcsin and arccos only ever output angles from a limited range: they're built to return one answer, not all the infinitely many true ones.