Inverse Trigonometric Functions
Questions here check principal value branches, domain–range restrictions, and the standard identities. Typical board items are a one-mark principal value, a simplification of a nested expression, and a 3-mark identity proof.
Principal branches
- sin⁻¹: [−1, 1] → [−π/2, π/2]
- cos⁻¹: [−1, 1] → [0, π]
- tan⁻¹: R → (−π/2, π/2)
- cot⁻¹: R → (0, π)
- sec⁻¹: |x| ≥ 1 → [0, π] except π/2; cosec⁻¹: |x| ≥ 1 → [−π/2, π/2] except 0.
Identities
- sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1, 1]; tan⁻¹x + cot⁻¹x = π/2 for all real x.
- 2 tan⁻¹x = tan⁻¹(2x/(1 − x²)) when |x| < 1.
- sin⁻¹(−x) = −sin⁻¹x, but cos⁻¹(−x) = π − cos⁻¹x.
Worked example
Evaluate sin⁻¹(sin 2π/3). The angle 2π/3 lies outside the principal range [−π/2, π/2]. Since sin(2π/3) = sin(π/3) and π/3 is inside the range, sin⁻¹(sin 2π/3) = π/3, not 2π/3.
Traps
- Cancelling an inverse function with its function without checking the branch - the classic error.
- Applying the cosine negative rule to sine.
- Ignoring domain restrictions when adding two inverse terms, e.g. tan⁻¹x + tan⁻¹y needs the product check before using the sum formula.