cos⁻¹ and its principal branch

Cosine is one-one on [0, π], where it falls from 1 to −1. Other intervals such as [−π, 0] and [π, 2π] also work, but [0, π] is the principal choice.

So cos⁻¹ : [−1, 1] → [0, π]. Its values are never negative.

cos⁻¹ is decreasing: cos⁻¹(−1) = π, cos⁻¹ 0 = π/2 and cos⁻¹ 1 = 0.

A negative input gives an obtuse angle, not a negative one. cos⁻¹(−1/2) = 2π/3, since cos 2π/3 = −1/2 and 2π/3 lies in [0, π].

cos⁻¹ is not odd. Instead cos⁻¹(−x) = π − cos⁻¹ x: cos⁻¹(−√3/2) = π − π/6 = 5π/6.

sin⁻¹ and cos⁻¹ share the domain [−1, 1] but have different ranges, [−π/2, π/2] and [0, π]; the two intervals overlap only on [0, π/2].