Graphs as mirror images

If (a, b) lies on the graph of y = sin x, then (b, a) lies on the graph of the inverse. Swapping the coordinates of every point reflects the graph in the line y = x.

So the graph of y = sin⁻¹ x is the principal piece of y = sin x, from (−π/2, −1) to (π/2, 1), mirrored in y = x. It runs from (−1, −π/2) to (1, π/2).

The mirrored full sine curve is a wavy vertical curve that fails the vertical-line test; that is the picture of why the inverse needs a branch. The principal branch is one arc of it.

The graph of y = cos⁻¹ x runs from (−1, π) down to (1, 0), passing through (0, π/2).

y = tan⁻¹ x is a gentle S through the origin between the horizontal asymptotes y = ±π/2; y = cot⁻¹ x falls from near π to near 0, crossing (0, π/2).

y = sec⁻¹ x and y = cosec⁻¹ x have two pieces each, one for x ≤ −1 and one for x ≥ 1, with a gap over (−1, 1). They approach y = π/2 and y = 0 respectively as |x| grows.