Inverse Trigonometric Functions

For inverse functions to exist

For inverse functions to exist, the original function must be one-to-one. Since sin x, cos x, tan x are periodic, we restrict their domains: sin⁻¹x ha

Step

Step-by-step approach: (1) Identify the function and its range, (2) Find the reference angle, (3) Determine the correct quadrant based on the range, (

Key identities

Key identities: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, sec⁻¹x + cosec⁻¹x = π/2. These arise because sin and cos are complementary functions. Ex

For tan⁻¹

For tan⁻¹: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy)) when xy < 1. For sin⁻¹: sin⁻¹x + sin⁻¹y = sin⁻¹[x√(1-y²) + y√(1-x²)]. Example: tan⁻¹(1/2) + tan⁻¹(1/3

When evaluating expressions like sin(cos⁻¹x)

When evaluating expressions like sin(cos⁻¹x) or tan(sin⁻¹x), use right triangles or algebraic identities. Method: (1) Let cos⁻¹x = θ, so cos θ = x, (2