Inverse Trigonometric Functions
What is tested
Inverse trigonometric functions test your grip on principal value branches and on the addition and substitution identities used to simplify expressions into a single principal value.
Branches to memorise
- sin⁻¹: [−π/2, π/2]; cos⁻¹: [0, π]; tan⁻¹: (−π/2, π/2); cot⁻¹: (0, π).
- sec⁻¹: [0, π] excluding π/2; cosec⁻¹: [−π/2, π/2] excluding 0.
- 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; when xy > 1 and x, y > 0, add π.
- 2tan⁻¹x = tan⁻¹[2x/(1−x²)] for |x| < 1.
Worked example
Simplify tan⁻¹(1/2) + tan⁻¹(1/3). Here xy = 1/6 < 1, so the plain formula applies: (1/2 + 1/3)/(1 − 1/6) = (5/6)/(5/6) = 1. The sum is tan⁻¹(1) = π/4.
Second check: sin⁻¹(sin 5π/6). Since 5π/6 is outside [−π/2, π/2], rewrite sin 5π/6 = sin(π − 5π/6) = sin(π/6); π/6 lies in the branch, so the value is π/6, not 5π/6.
Marks-losing traps
Using the tan⁻¹ addition formula when xy > 1 without the π correction; cancelling sin⁻¹ with sin outside the principal branch; reading sin⁻¹x as 1/sin x; forgetting sec⁻¹ and cosec⁻¹ are undefined at 0; mixing degrees and radians in a principal-value answer.