Chapter 2 — Inverse Trigonometric Functions

Chapter 2 of the Class 12 Maths NCERT textbook, "Inverse Trigonometric Functions", covers 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.

• Why inverses need restricted domains — Because trigonometric functions repeat their values, they cannot be reversed as they stand. The chapter's core idea is to trim each function's domain to a single one-one stretch — the principal value branch — so a genuine inverse can exist.
• Reading inverse functions from graphs — Each inverse trig graph is the mirror image of the original across the line y = x, which makes the swap of input and output visual. This geometric view explains why the range of the inverse is exactly the restricted domain chosen for the original.
• Simplifying with substitution — Rather than manipulate inverse trig expressions directly, the chapter teaches substituting x with sinθ, cosθ or tanθ to fold complicated forms into familiar double-angle or sum identities, turning awkward simplifications into routine trigonometry.

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.