Chapter 2 — Inverse Trigonometric Functions

NCERT Class 12 Maths Chapter 2 covers Inverse Trigonometric Functions — the restrictions on domains and ranges of trigonometric functions that make their inverses well-defined, along with principal value branches and key properties used in calculus and engineering.

Chapter 2 of NCERT Class 12 Mathematics (Part I) introduces Inverse Trigonometric Functions by first establishing why restrictions on the natural domains of trigonometric functions are necessary for their inverses to exist. Each inverse function — sin⁻¹, cos⁻¹, tan⁻¹, cot⁻¹, sec⁻¹, and cosec⁻¹ — is defined with its principal value branch, domain, and range. The chapter covers graphical representations (obtained by reflecting y = f(x) across y = x), important properties such as sin(sin⁻¹x) = x and sin⁻¹(sin x) = x within appropriate domains, and simplification techniques using substitution. These functions are foundational in calculus for defining integrals and have wide applications in science and engineering.

1. 01Inverse trigonometric functions exist only when the domain of the original function is restricted to make it one-one and onto; the chosen restricted interval is called the principal value branch.
2. 02Principal value branch domains and ranges: 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: ℝ−(−1,1) → [0,π]−{π/2}; cosec⁻¹: ℝ−(−1,1) → [−π/2, π/2]−{0}.
3. 03sin⁻¹x must not be confused with (sin x)⁻¹ = 1/sin x; the superscript −1 on an inverse trig function denotes the inverse function, not a reciprocal.
4. 04The graph of y = sin⁻¹x (or any inverse trig function) is the reflection of the original graph across the line y = x; the principal value branch is highlighted as the standard output.
5. 05Key identities hold within principal value domains: sin(sin⁻¹x) = x for x ∈ [−1,1] and sin⁻¹(sinx) = x for x ∈ [−π/2, π/2], with analogous results for all six functions.
6. 06Complex expressions involving inverse trig functions can be simplified by substituting x = sinθ, x = cosθ, or x = tanθ to convert them into simpler forms such as double-angle or sum formulas.