The six principal branches

sin⁻¹ : [−1, 1] → [−π/2, π/2]. cos⁻¹ : [−1, 1] → [0, π].

cosec⁻¹ : R − (−1, 1) → [−π/2, π/2] − {0}. sec⁻¹ : R − (−1, 1) → [0, π] − {π/2}.

tan⁻¹ : R → (−π/2, π/2). cot⁻¹ : R → (0, π).

A pattern to remember them by: sin⁻¹, tan⁻¹ and cosec⁻¹ live around 0, in the right half of the circle from −π/2 to π/2; cos⁻¹, cot⁻¹ and sec⁻¹ live in the upper half, from 0 to π.

So for a negative input, the first group returns a negative angle (a fourth-quadrant angle measured clockwise) and the second group returns an angle between π/2 and π.

The ranges lose their endpoints exactly where the original function blows up: tan at ±π/2, cot at 0 and π, cosec at 0, sec at π/2.