Inverse trigonometric functions

The essentials

The trigonometric functions are not one-one on R, so they have no inverse on their full domains. By restricting each one to an interval where it is one-one and onto its range, we obtain an inverse. The chosen interval gives the principal value branch, and the output of the inverse function is called the principal value.

For example, sin x is a bijection from [−π/2, π/2] onto [−1, 1], so sin⁻¹ : [−1, 1] → [−π/2, π/2]. The graph of an inverse function is the mirror image of the restricted original graph in the line y = x.

Key points

• sin⁻¹x: domain [−1, 1], range [−π/2, π/2].
• cos⁻¹x: domain [−1, 1], range [0, π].
• tan⁻¹x: domain R, range (−π/2, π/2).
• cot⁻¹x: domain R, range (0, π).
• sec⁻¹x: domain |x| ≥ 1, range [0, π] except π/2.
• cosec⁻¹x: domain |x| ≥ 1, range [−π/2, π/2] except 0.
• sin⁻¹(−x) = −sin⁻¹x and tan⁻¹(−x) = −tan⁻¹x, but cos⁻¹(−x) = π − cos⁻¹x.
• sin⁻¹x + cos⁻¹x = π/2 for x in [−1, 1].
• tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)] when xy < 1.

Exam tips

• sin⁻¹(sin θ) = θ only when θ already lies in [−π/2, π/2]; otherwise find the angle in the principal range with the same sine.
• Remember that sin⁻¹x is not the same as 1/sin x.

Common mistakes

• Giving cos⁻¹(−½) as −π/3. Values of cos⁻¹ are never negative; the answer is 2π/3.
• Using the tan⁻¹ addition formula without checking whether xy < 1.