NCERT Solutions for Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions

Short Note:

• sin⁻¹x: domain [-1, 1], range [-π/2, π/2]; cos⁻¹x: domain [-1, 1], range [0, π]; tan⁻¹x: domain R, range (-π/2, π/2).

• cosec⁻¹x and sec⁻¹x are defined for |x| ≥ 1, while cot⁻¹x is defined for all real x, with range (0, π).

Short Note:

• sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; sec⁻¹x + cosec⁻¹x = π/2, each valid within its respective domain.

• sin⁻¹(-x) = -sin⁻¹x and tan⁻¹(-x) = -tan⁻¹x (odd functions), whereas cos⁻¹(-x) = π - cos⁻¹x.

Short Note:

• Inverse trigonometric functions are obtained by restricting trigonometric functions to intervals where they become one-one and onto, ensuring a well-defined inverse.

• Composite expressions such as sin(cos⁻¹x) or tan(sin⁻¹x) can be simplified using right-triangle relationships or the Pythagorean identity.