Inverse Trigonometric Functions — Class 12

Introduction

This page gives a complete, exam-ready summary of Inverse Trigonometric Functions. Study the definitions, principal-value ranges, identities, derivatives, integrals, geometric diagrams, solved examples and practice questions — all designed for long-term retention and quick revision.

1. Definitions & Principal Values

1.1 What is an inverse trigonometric function?

Given a trig function y = sin x (or cos x, tan x...), its inverse is the function that returns the angle for a given value: x = arcsin y, x = arccos y, x = arctan y, etc. Because trig functions are not one-to-one on ℝ, we restrict to principal ranges to define inverses.

1.2 Principal-value ranges (standard)

• y = arcsin x: domain x ∈ [−1,1], range y ∈ [−π/2, π/2]
• y = arccos x: domain x ∈ [−1,1], range y ∈ [0, π]
• y = arctan x: domain x ∈ ℝ, range y ∈ (−π/2, π/2)
• y = arccot x: range y ∈ (0, π)
• y = arcsec x: domain |x| ≥ 1, range y ∈ [0, π] \ {π/2}
• y = arccsc x: domain |x| ≥ 1, range y ∈ [−π/2, π/2] \ {0}

Mapping / Unit-circle diagram (visual)

Replace with high-contrast SVG for best clarity. Example: show unit circle, mark angle θ, show sinθ on y-axis and label arcsin(sinθ) principal branch.

2. Identities & Useful Relations

Use the following to convert between trig & inverse trig, and to simplify expressions.

2.1 Basic identities

• sin(arcsin x) = x for x ∈ [−1,1]
• arcsin(sin x) = x only if x ∈ [−π/2, π/2]
• cos(arccos x) = x, arctan(tan x) = x on their principal ranges.

2.2 Composition & complementary relations

• arcsin x + arccos x = π/2 for x ∈ [−1,1]
• arctan x + arccot x = π/2 (for x>0, careful with branches)
• For algebraic simplification: use triangle method — put x = sin θ and use right-triangle relations to express √(1−x²).

3. Derivatives (Important)

These derivatives hold on interior of domains where functions are defined.

• d/dx[arcsin x] = 1 / √(1 - x²), |x|<1
• d/dx[arccos x] = −1 / √(1 - x²), |x|<1
• d/dx[arctan x] = 1 / (1 + x²), x∈ℝ
• d/dx[arccot x] = −1 / (1 + x²), x∈ℝ
• d/dx[arcsec x] = 1 / (|x|√(x²−1)), |x|>1
• d/dx[arccsc x] = −1 / (|x|√(x²−1)), |x|>1

4. Standard Integrals involving inverse trig

• ∫ dx / √(1 − x²) = arcsin x + C
• ∫ dx / (1 + x²) = arctan x + C
• ∫ dx / (|x|√(x² − 1)) = arcsec |x| + C
• Integrals reducing to inverse trig often use substitution x = sin θ, or partial-fraction/trig-sub techniques.

5. Solved Examples (step-by-step)

Example 1

Problem: Differentiate y = arcsin(2x).

Solution: dy/dx = 1/√(1−(2x)²) · d/dx(2x) = 2 / √(1−4x²).

Example 2

Problem: Evaluate ∫ dx/(1 + x²).

Solution: Standard integral = arctan x + C.

Example 3

Problem: Solve for x: arctan x + arctan 2 = π/4.

Solution idea: Use tan(A+B) formula: tan(arctan x + arctan 2) = (x+2)/(1−2x). Set equal to tan(π/4)=1. Solve (x+2)/(1−2x) = 1 → x+2 = 1−2x → 3x = −1 → x = −1/3. Check branch validity.

Include more examples: inverse trig composition, domain checks, branch adjustments, composite differentiation (chain rule), and integrals by substitution.

6. Summary Table

7. Practice Questions (High-yield)

8. FAQ