Learning Objectives
- Understand inverse trigonometric functions and their domains/ranges
- Learn principal value branches
- Study important properties and identities
- Solve problems using inverse trigonometric formulas
Key Concepts
Principal Value Branches
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). cot-1x: Domain R, Range (0, π). sec-1x: Domain R-(-1,1), Range [0,π]-{π/2}. cosec-1x: Domain R-(-1,1), Range [-π/2, π/2]-{0}.
Key Properties
Negative arguments: sin-1(-x) = -sin-1x. cos-1(-x) = π - cos-1x. tan-1(-x) = -tan-1x. cosec-1(-x) = -cosec-1x. sec-1(-x) = π - sec-1x. cot-1(-x) = π - cot-1x.
Reciprocal relations: sin-1(1/x) = cosec-1x. cos-1(1/x) = sec-1x. tan-1(1/x) = cot-1x (for x > 0).
Complementary relations: sin-1x + cos-1x = π/2. tan-1x + cot-1x = π/2. sec-1x + cosec-1x = π/2.
Important Formulas
tan-1x + tan-1y = tan-1[(x+y)/(1-xy)] when xy < 1.
tan-1x - tan-1y = tan-1[(x-y)/(1+xy)] when xy > -1.
2tan-1x = sin-1[2x/(1+x2)] for |x| ≤ 1.
2tan-1x = cos-1[(1-x2)/(1+x2)] for x ≥ 0.
2tan-1x = tan-1[2x/(1-x2)] for |x| < 1.
3tan-1x = tan-1[(3x-x3)/(1-3x2)].
Conversion Between Inverse Trig Functions
sin-1x = cos-1√(1-x2) = tan-1[x/√(1-x2)] for 0 < x < 1. These conversions are essential for simplification problems in JEE.
Summary
Inverse trigonometric functions are defined by restricting domains of trig functions to make them bijective. Each has a specific principal value branch. Key properties include complementary relations, negative argument rules, and addition formulas for tan-1.
Important Terms
- Principal value: The unique value in the defined range of the inverse function
- Domain restriction: Limiting input to make inverse well-defined
- Complementary pair: sin-1x + cos-1x = π/2
Quick Revision
- sin-1x ∈ [-π/2, π/2]; cos-1x ∈ [0, π]; tan-1x ∈ (-π/2, π/2)
- sin-1x + cos-1x = π/2; tan-1x + cot-1x = π/2
- sin-1(-x) = -sin-1x (odd); cos-1(-x) = π - cos-1x
- tan-1x + tan-1y = tan-1[(x+y)/(1-xy)] if xy < 1
- 2tan-1x = sin-1(2x/(1+x2)) for |x| ≤ 1
- tan-1(1) = π/4; sin-1(1) = π/2; cos-1(1) = 0