Inverse Trigonometric Functions
Learn Inverse Trigonometric Functions with notes, examples, and practice questions.
Chapter Overview
Principal value branch keeps the inverse function single-valued — this is what CBSE means by sin⁻¹x, not the full infinite family.
Inverse trigonometric functions reverse the trigonometric functions — but only on restricted domains so they become one-one and onto.
You must know principal value branches, domains, ranges, and the identities that simplify expressions like tan⁻¹x + cot⁻¹x.
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 (−∞,−1]∪[1,∞), Range [0,π]\{π/2}
cosec⁻¹x
Domain (−∞,−1]∪[1,∞), Range [−π/2,π/2]\{0}
Key Properties
- sin(sin⁻¹x) = x for x ∈ [−1,1]
- sin⁻¹(sin x) = x only when x is in principal range
- sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1,1]
- tan⁻¹x + cot⁻¹x = π/2 for x ∈ ℝ
- 2 tan⁻¹x = sin⁻¹(2x/(1+x²)) for |x| ≤ 1
Differentiation (used with chain rule)
d/dx sin⁻¹x
= 1/√(1−x²)
d/dx cos⁻¹x
= −1/√(1−x²)
d/dx tan⁻¹x
= 1/(1+x²)
d/dx cot⁻¹x
= −1/(1+x²)
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Principal Value
Find sin⁻¹(−1/2).
- 1sin θ = −1/2 in [−π/2, π/2]
- 2θ = −π/6
Answer
−π/6
Example 2: Identity
Simplify tan⁻¹(1/2) + tan⁻¹(1/3).
- 1Use tan⁻¹a + tan⁻¹b = tan⁻¹((a+b)/(1−ab)) when ab < 1
- 2=(1/2+1/3)/(1−1/6) = (5/6)/(5/6) = 1
- 3tan⁻¹(1) = π/4
Answer
π/4
Example 3: Expression
If sin⁻¹x + sin⁻¹y = π/2, find cos⁻¹x + cos⁻¹y.
- 1sin⁻¹x + cos⁻¹x = π/2
- 2So sin⁻¹y = cos⁻¹x
- 3Similarly cos⁻¹y = sin⁻¹x
- 4Sum = π/2
Answer
π/2
Key Points to Remember
- ✓ Always check domain before applying identities
- ✓ Principal value ranges are exam-critical
- ✓ Convert to tan⁻¹ for addition formulas when stuck
Exam Tips
- • Draw unit circle for principal value questions
- • Memorise sin⁻¹x + cos⁻¹x = π/2
- • For proofs, convert all terms to same inverse function
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Inverse Trig Identities
- sin⁻¹x + cos⁻¹x = π/2, x ∈ [−1,1]
- tan⁻¹x + cot⁻¹x = π/2, x ∈ ℝ
- sin⁻¹(−x) = −sin⁻¹x ; tan⁻¹(−x) = −tan⁻¹x
- cos⁻¹(−x) = π − cos⁻¹x ; cot⁻¹(−x) = π − cot⁻¹x
- sin⁻¹x = tan⁻¹(x/√(1−x²))
- cos⁻¹x = tan⁻¹(√(1−x²)/x)
Practice MCQs — Inverse Trigonometric Functions
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Principal value of cos⁻¹(−1/2) is: