Class 12 Maths Notes: Inverse Trigonometric Functions (Chapter 2)
Master Chapter 2 of Class 12 Mathematics. Understand principal value branches, domain and range, key identities with conditions, and common exam errors in inverse trig for board exams.
Students assume that sin⁻¹(sin x) = x always. This is wrong. Trigonometric functions are not one-to-one over their full domain. For an inverse to exist, the domain must be restricted to make the function one-to-one.
sin⁻¹(sin x) = x only when x ∈ [−π/2, π/2]. Outside this range, you must first find the equivalent angle inside the principal value range.
2. Principal Value Branches
| Function | Domain | Principal Value Range |
|---|---|---|
| sin⁻¹ x | [−1, 1] | [−π/2, π/2] |
| cos⁻¹ x | [−1, 1] | [0, π] |
| tan⁻¹ x | ℝ | (−π/2, π/2) |
| cot⁻¹ x | ℝ | (0, π) |
| sec⁻¹ x | |x| ≥ 1 | [0, π] − {π/2} |
| cosec⁻¹ x | |x| ≥ 1 | [−π/2, π/2] − {0} |
⚠️ Watch Out! — Always Check the Range First
Evaluate sin⁻¹(sin 2π/3) step by step:
Step 1: Is 2π/3 in [−π/2, π/2]? No.
Step 2: sin(2π/3) = sin(π − π/3) = sin(π/3) = √3/2
Step 3: sin⁻¹(√3/2) = π/3. The answer is π/3, not 2π/3.
3. Key Identities — With Their Conditions
Essential Identities (Memorise With Conditions)
- sin⁻¹ x + cos⁻¹ x = π/2 when x ∈ [−1, 1]
- tan⁻¹ x + cot⁻¹ x = π/2 when x ∈ ℝ
- sec⁻¹ x + cosec⁻¹ x = π/2 when |x| ≥ 1
- tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x+y)/(1−xy)] when xy < 1
- tan⁻¹ x + tan⁻¹ y = π + tan⁻¹[(x+y)/(1−xy)] when xy > 1, x > 0
- 2 tan⁻¹ x = sin⁻¹(2x/(1+x²)) when |x| ≤ 1
4. Proof Structure: Simplification Problems
Questions ask you to simplify expressions like tan⁻¹[(cos x − sin x)/(cos x + sin x)].
Standard approach: Divide numerator and denominator by cos x → convert to tan form → use addition formula:
tan⁻¹[(1 − tan x)/(1 + tan x)] = tan⁻¹[tan(π/4 − x)] = π/4 − x
Students who practise this conversion recognise it quickly. Students who do not treat each question as entirely new.
Summary: Key Facts at a Glance
| Concept | Key Rule |
|---|---|
| sin⁻¹(sin x) | = x only if x ∈ [−π/2, π/2] |
| sec⁻¹ domain | |x| ≥ 1 (excludes (−1,1)) |
| tan⁻¹ + tan⁻¹ addition | Condition: check sign of xy vs 1 |
| cos⁻¹ range | [0, π] — always non-negative |
Practice Questions (PYQs)
- Find the principal value of sin⁻¹(sin 5π/6).
- Prove that tan⁻¹(1/2) + tan⁻¹(1/3) = π/4.
- Write the domain and range of f(x) = cos⁻¹ x. Why is the range [0, π] and not [−π/2, π/2]?
- Simplify: tan⁻¹[(1 + x)/(1 − x)] for x < 1.
- Prove that sin⁻¹(3/5) − cos⁻¹(12/13) = sin⁻¹(16/65).