sin⁻¹ and its principal branch

Restricted to [−π/2, π/2], sine is one-one and onto [−1, 1]. Its inverse is sin⁻¹ : [−1, 1] → [−π/2, π/2], also written arc sin.

y = sin⁻¹ x means: y is the angle in [−π/2, π/2] whose sine is x. So sin⁻¹(1/2) = π/6 and not 5π/6, even though sin 5π/6 = 1/2 too, because 5π/6 lies outside [−π/2, π/2].

sin⁻¹ is an increasing function. It runs from sin⁻¹(−1) = −π/2 through sin⁻¹ 0 = 0 to sin⁻¹ 1 = π/2.

It is odd: sin⁻¹(−x) = −sin⁻¹ x, because the principal interval is symmetric about 0. So sin⁻¹(−1/2) = −π/6.

sin⁻¹ x is undefined for |x| > 1: no angle has a sine of 2.

On another branch the answers shift. On the branch with range [π/2, 3π/2], the angle whose sine is 1/2 is 5π/6; on [−3π/2, −π/2] it is −7π/6. These are not principal values.

The −1 is part of the name, not a power: sin⁻¹ x is an angle, while (sin x)⁻¹ = 1/sin x = cosec x. The same holds for every trigonometric function.