Inverse Trigonometric Functions
Inverse Trigonometric Functions tests the principal-value branches and the identities that let you combine inverse sine, cosine and tangent expressions. Markers are strict about the range of the answer, so the branch matters as much as the value.
Principal value branches
- sin⁻¹: domain [−1, 1], range [−π/2, π/2].
- cos⁻¹: domain [−1, 1], range [0, π].
- tan⁻¹: R → (−π/2, π/2); cot⁻¹: R → (0, π).
- sec⁻¹: |x| ≥ 1 → [0, π] − {π/2}; cosec⁻¹: |x| ≥ 1 → [−π/2, π/2] − {0}.
Identities
- sin⁻¹(−x) = −sin⁻¹x, tan⁻¹(−x) = −tan⁻¹x, but cos⁻¹(−x) = π − cos⁻¹x and cot⁻¹(−x) = π − cot⁻¹x.
- sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; sec⁻¹x + cosec⁻¹x = π/2.
- tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 − xy)) when xy < 1; add π when xy > 1 with x, y > 0, and subtract π when xy > 1 with x, y < 0.
- 2tan⁻¹x = tan⁻¹(2x/(1 − x²)) for |x| < 1.
Worked example
Evaluate tan⁻¹1 + tan⁻¹2 + tan⁻¹3. Take the first pair: x = 1, y = 2, so xy = 2 > 1 and both are positive. Therefore tan⁻¹1 + tan⁻¹2 = π + tan⁻¹((1 + 2)/(1 − 2)) = π + tan⁻¹(−3) = π − tan⁻¹3. Adding tan⁻¹3 gives π − tan⁻¹3 + tan⁻¹3 = π.
Traps
- Writing sin⁻¹(sin x) = x for all x; it holds only on the principal branch, so sin⁻¹(sin 2π/3) = π/3, not 2π/3.
- Ignoring the xy < 1 condition in the tangent addition formula.
- Reporting a value outside the principal range, e.g. giving 7π/6 instead of 5π/6 for cos⁻¹(cos 7π/6).
- Forgetting that sin⁻¹2 is undefined.