Inverse Trigonometric Functions
Board: CBSE | Class: Class 12
Comprehensive study notes for Inverse Trigonometric Functions by Ajay Yadav.
Key Concepts
Principal Values
Inverse trig functions have restricted domains to make them one-one. sin⁻¹x: domain [-1,1], range (-&pi/2,&pi/2). cos⁻¹x: domain [-1,1], range (0,&pi). tan⁻¹x: domain R, range (-π/2,π/2).
Properties
sin⁻¹(sinθ) = θ if θ ∈ (-&pi/2,&pi/2). sin(sin⁻¹x) = x if x ∈ [-1,1]. Similarly for other functions.
Identities
sin⁻¹x + cos⁻¹x = π/2. tan⁻¹x + cot⁻¹x = π/2. sec⁻¹x + cosec⁻¹x = π/2.
tan⁻¹ Formulas
tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] if xy<1. tan⁻¹x – tan⁻¹y = tan⁻¹[(x-y)/(1+xy)]. 2tan⁻¹x = tan⁻¹(2x/(1-x²)) = sin⁻¹(2x/(1+x²)) = cos⁻¹((1-x²)/(1+x²)).
Important Formulas
| sin⁻¹x range | (-&pi/2,&pi/2) |
| cos⁻¹x range | (0,&pi) |
| Sum identity | sin⁻¹x + cos⁻¹x = π/2 |
| 2tan⁻¹x | tan⁻¹(2x/(1-x²)) |
Solved Examples
Example 1: Find principal value of sin⁻¹(1/2).
Solution: sin⁻¹(1/2) = π/6 (since sin(π/6)=1/2 and π/6 ∈ (-&pi/2,&pi/2)).
Example 2: Evaluate: tan⁻¹(1) + cos⁻¹(0).
Solution: tan⁻¹(1)=π/4, cos⁻¹(0)=π/2. Sum = 3π/4.
Example 3: Simplify: sin⁻¹(sin(5π/6)).
Solution: sin(5π/6)=1/2. sin⁻¹(1/2)=π/6 (principal value).
Practice Questions
- Find principal value of cos⁻¹(-1/2).
- Prove: tan⁻¹1 + tan⁻¹2 + tan⁻¹3 = π.
- Simplify: 2tan⁻¹(1/3).
- Find the domain of sin⁻¹x + cos⁻¹x.
- Evaluate: tan⁻¹(tan(3π/4)).
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Video Lessons
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