Inverse Trigonometric Functions — Important Questions

Domain is [-1, 1], Range is [-π/2, π/2]

For sin⁻¹x (principal branch): Domain = [-1, 1] because sine values lie between -1 and 1. Range = [-π/2, π/2] which is the principal value branch. The function is not defined for values outside [-1, 1].

π/2

We need to find the angle whose cosine is 0. From trigonometric values, cos(π/2) = 0. Since π/2 lies in the principal value range [0, π] for cos⁻¹, the answer is π/2.

π/4

tan⁻¹(1) = π/4 (since tan(π/4) = 1) and tan⁻¹(0) = 0 (since tan(0) = 0). Therefore, tan⁻¹(1) + tan⁻¹(0) = π/4 + 0 = π/4.

0, π/4, π

The range of sec⁻¹x is [0, π] - {π/2}. This means all values from 0 to π are included except π/2. Therefore, 0, π/4, and π are in the range, but π/2 and 3π/2 are not.