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6.Properties & Graphs
Class 12 • Mathematics • NCERT
Inverse Trigonometric Functions – Properties & Graphs
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This lesson covers the standard properties and graphs of inverse trigonometric functions as prescribed by NCERT Class 12. These results are essential for simplification, proof-based questions, and graph interpretation.
Lesson Objectives
- Learn standard NCERT properties.
- Understand inverse trig graphs conceptually.
- Use properties for simplification.
- Prepare for board-style questions.
1. Fundamental Properties (NCERT)
sin(sin⁻¹ x) = x, |x| ≤ 1
cos(cos⁻¹ x) = x, |x| ≤ 1
tan(tan⁻¹ x) = x, x ∈ ℝ
cos(cos⁻¹ x) = x, |x| ≤ 1
tan(tan⁻¹ x) = x, x ∈ ℝ
2. Trigonometric of Inverse (Very Important)
sin⁻¹(sin x) = x, x ∈ [−π/2, π/2]
cos⁻¹(cos x) = x, x ∈ [0, π]
tan⁻¹(tan x) = x, x ∈ (−π/2, π/2)
⚠️ Outside these intervals, answers change — this is a common exam trap.
3. Complementary Angle Properties
sin⁻¹ x + cos⁻¹ x = π/2, |x| ≤ 1
tan⁻¹ x + cot⁻¹ x = π/2, x > 0
4. Graphs of Inverse Trigonometric Functions
Graphs of inverse trigonometric functions are obtained by reflecting the restricted graphs of trigonometric functions about the line y = x.
• y = sin⁻¹ x → increasing curve from (−1, −π/2) to (1, π/2)
• y = cos⁻¹ x → decreasing curve from (−1, π) to (1, 0)
• y = tan⁻¹ x → increasing curve with horizontal asymptotes
• y = cos⁻¹ x → decreasing curve from (−1, π) to (1, 0)
• y = tan⁻¹ x → increasing curve with horizontal asymptotes
5. Important NCERT Notes
• Inverse trig graphs are one-one
• Domains are restricted intentionally
• Graph-based questions test range understanding
• Domains are restricted intentionally
• Graph-based questions test range understanding
Practice Questions (NCERT)
- State sin(sin⁻¹ x).
- Find cos⁻¹(cos π/3).
- Find sin⁻¹(sin π/6).
- Evaluate tan⁻¹(tan π/4).
- State the range of sin⁻¹ x.
- State the range of cos⁻¹ x.
- What is the nature of graph of y = sin⁻¹ x?
- Why are inverse trig graphs one-one?
- State sin⁻¹ x + cos⁻¹ x.
- Is this topic part of NCERT Class 12 syllabus?
✅ Show Answer Key
- x
- π/3
- π/6
- π/4
- [−π/2, π/2]
- [0, π]
- Increasing
- Restricted domain
- π/2
- Yes
© Aviate Learning – Inverse Trigonometric Functions (NCERT Class 12)