Relations, Functions and Inverse Trigonometric Functions
Relations, Functions and Inverse Trigonometric Functions
This domain tests whether you can classify relations, decide one-one and onto behaviour, compose and invert functions, and evaluate inverse trigonometric expressions using principal branches.
Core rules
- A relation R on A is reflexive, symmetric and transitive; an equivalence relation splits A into disjoint classes.
- A function f:A→B is one-one if f(x₁)=f(x₂) forces x₁=x₂, and onto if every y∈B has a preimage. Only bijections have inverses.
- Principal ranges: sin⁻¹, cosec⁻¹ ⊂ [−π/2, π/2] (cosec excludes 0); cos⁻¹, sec⁻¹ ⊂ [0, π] (sec excludes π/2); tan⁻¹ ⊂ (−π/2, π/2); cot⁻¹ ⊂ (0, π).
- Standard identities: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, sec⁻¹x + cosec⁻¹x = π/2; sin⁻¹(−x) = −sin⁻¹x but cos⁻¹(−x) = π − cos⁻¹x.
Worked example
Evaluate sin⁻¹(sin 5π/6).
5π/6 lies outside [−π/2, π/2], so you may not return 5π/6. First sin(5π/6) = 1/2. Then sin⁻¹(1/2) = π/6 because π/6 is the unique angle in [−π/2, π/2] with sine 1/2. Answer: π/6.
Traps
- Forgetting that sin⁻¹(sin x) = x only when x is in the principal interval.
- Counting relations: A×A has n² pairs, so there are 2^(n²) relations and 2^(n²−n) reflexive relations.
- Confusing range of f with domain of f⁻¹; invertible means bijective on the stated domain.
- Writing cos⁻¹(−1/2) = 2π/3, not −π/3.