Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Overview
Inverse trigonometric functions reverse the trig ratios. Because sin, cos, tan are many-to-one, each inverse is defined on a restricted principal value range.
1. Domains and Principal Ranges
sin⁻¹x : domain [−1, 1] range [−π/2, π/2]
cos⁻¹x : domain [−1, 1] range [0, π]
tan⁻¹x : domain (−∞, ∞) range (−π/2, π/2)
cot⁻¹x : domain (−∞, ∞) range (0, π)
sec⁻¹x : domain |x| ≥ 1 range [0, π] − {π/2}
cosec⁻¹x: domain |x| ≥ 1 range [−π/2, π/2] − {0}
2. Key Identities
sin⁻¹x + cos⁻¹x = π/2
tan⁻¹x + cot⁻¹x = π/2
sec⁻¹x + cosec⁻¹x = π/2
tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)] (xy < 1)
2 tan⁻¹x = tan⁻¹[2x/(1−x²)] = sin⁻¹[2x/(1+x²)]
Worked Examples
Example 1 — Principal value
Find the principal value of sin⁻¹(1/2).
sin θ = 1/2, θ ∈ [−π/2, π/2] → θ = π/6
Example 2 — cos⁻¹ of a negative
Find cos⁻¹(−1/2).
cos θ = −1/2, θ ∈ [0, π] → θ = 2π/3
Example 3 — Using complementary identity
Evaluate sin⁻¹(1/2) + cos⁻¹(1/2).
= π/2 (since sin⁻¹x + cos⁻¹x = π/2)
Example 4 — Sum of two tan⁻¹
Evaluate tan⁻¹1 + tan⁻¹2 + tan⁻¹3.
tan⁻¹1 + tan⁻¹2 = tan⁻¹[(1+2)/(1−2)] = tan⁻¹(−3) = π − tan⁻¹3
Adding tan⁻¹3: π − tan⁻¹3 + tan⁻¹3 = π
Example 5 — Double angle form
Write 2 tan⁻¹(1/3) as a single inverse.
= tan⁻¹[2(1/3)/(1 − 1/9)] = tan⁻¹[(2/3)/(8/9)] = tan⁻¹(3/4)
Common Mistakes
- Giving an answer outside the principal range.
- Forgetting cos⁻¹(negative) lies in (π/2, π].
- Using the tan⁻¹ sum formula when xy > 1 (needs an adjustment of π).
- Confusing domain restrictions of sec⁻¹/cosec⁻¹.
Exam Tips & Revision Checklist
- Memorise the principal value ranges — most marks hinge on them.
- sin⁻¹x + cos⁻¹x = π/2 simplifies many problems instantly.
- Practised: principal values, negative arguments, complementary identity, tan⁻¹ sums, double angle.
Frequently Asked Questions
sin⁻¹x has domain [−1, 1] and principal value range [−π/2, π/2].