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5.Inverse Trigonometric Functions
Class 12 • Mathematics • NCERT
Inverse Trigonometric Functions
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This lesson introduces inverse trigonometric functions as prescribed by NCERT Class 12. Students learn the meaning, domains, ranges, and principal values required to solve exam-oriented problems accurately.
Lesson Objectives
- Understand inverse trigonometric functions.
- Learn principal value branches.
- Identify correct domain and range.
- Solve NCERT exam-type questions.
1. Meaning of Inverse Trigonometric Functions
Inverse trigonometric functions are inverse of trigonometric functions, defined by restricting the domain of trigonometric functions to make them one-one.
If sin y = x, then sin⁻¹ x = y
2. Why Domain Restriction Is Needed
Trigonometric functions are not one-one over their entire domain. Hence, their domains are restricted to define inverses.
Example: sin x is restricted to [−π/2, π/2]
3. Principal Value Branches (NCERT)
sin⁻¹ x → Domain: [−1, 1], Range: [−π/2, π/2]
cos⁻¹ x → Domain: [−1, 1], Range: [0, π]
tan⁻¹ x → Domain: ℝ, Range: (−π/2, π/2)
4. Important NCERT Results
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
5. Simple NCERT Examples
sin⁻¹(1/2) = π/6
cos⁻¹(0) = π/2
tan⁻¹(1) = π/4
Practice Questions (NCERT)
- Why are inverse trigonometric functions defined?
- State the domain of sin⁻¹ x.
- State the range of cos⁻¹ x.
- Find sin⁻¹(−1).
- Find cos⁻¹(1).
- Find tan⁻¹(0).
- Is sin(sin⁻¹ x) = x always true?
- What is the range of tan⁻¹ x?
- Are inverse trigonometric functions one-one?
- Is this topic part of NCERT Class 12 syllabus?
✅ Show Answer Key
- To make trigonometric functions invertible
- [−1, 1]
- [0, π]
- −π/2
- 0
- 0
- Yes, within domain
- (−π/2, π/2)
- Yes
- Yes
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