📐 CHAPTER 2: INVERSE TRIGONOMETRIC FUNCTIONS
Subject: Mathematics | Class: XII | CBSE
📖 INTRODUCTION
Inverse Trigonometric Functions
Trigonometric functions are many-one functions. The inverse of a function exists only if the function is bijective.
By restricting the domain of trigonometric functions, they become bijective and their inverses are defined within the restricted domain.
The inverse of f is denoted by f⁻¹.
Example
Let y = f(x) = sin x, then its inverse is x = sin⁻¹ y.
💡 Note: sin⁻¹ x should not be confused with (sin x)⁻¹ = 1/sin x or sin(x⁻¹). Similarly for other trigonometric functions.
📋 DOMAIN AND RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
🔹 Principal Value Branch
The value of an inverse trigonometric function which lies in the range of principal value branch is called the principal value of the inverse trigonometric function.
Note: Whenever no branch is mentioned, we consider the principal value branch.
| Function | Domain | Range (Principal Value Branch) |
|---|---|---|
| sin⁻¹ x | [-1, 1] | [-π/2, π/2] |
| cos⁻¹ x | [-1, 1] | [0, π] |
| tan⁻¹ x | R | (-π/2, π/2) |
| cosec⁻¹ x | (-∞, -1] ∪ [1, ∞) | [-π/2, π/2] - {0} |
| sec⁻¹ x | (-∞, -1] ∪ [1, ∞) | [0, π] - {π/2} |
| cot⁻¹ x | R | (0, π) |
📜 IMPORTANT FORMULAS
🔹 sin⁻¹(sin θ) and sin(sin⁻¹ x)
- sin⁻¹(sin θ) = θ, ∀ θ ∈ [-π/2, π/2]
- sin(sin⁻¹ x) = x, ∀ x ∈ [-1, 1]
🔹 cos⁻¹(cos θ) and cos(cos⁻¹ x)
- cos⁻¹(cos θ) = θ, ∀ θ ∈ [0, π]
- cos(cos⁻¹ x) = x, ∀ x ∈ [-1, 1]
🔹 tan⁻¹(tan θ) and tan(tan⁻¹ x)
- tan⁻¹(tan θ) = θ, ∀ θ ∈ (-π/2, π/2)
- tan(tan⁻¹ x) = x, ∀ x ∈ R
🔹 cosec⁻¹(cosec θ) and cosec(cosec⁻¹ x)
- cosec⁻¹(cosec θ) = θ, ∀ θ ∈ [-π/2, π/2] - {0}
- cosec(cosec⁻¹ x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)
🔹 sec⁻¹(sec θ) and sec(sec⁻¹ x)
- sec⁻¹(sec θ) = θ, ∀ θ ∈ [0, π] - {π/2}
- sec(sec⁻¹ x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)
🔹 cot⁻¹(cot θ) and cot(cot⁻¹ x)
- cot⁻¹(cot θ) = θ, ∀ θ ∈ (0, π)
- cot(cot⁻¹ x) = x, ∀ x ∈ R
📐 PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS
🔹 Negative Arguments
- sin⁻¹(-x) = -sin⁻¹ x
- cos⁻¹(-x) = π - cos⁻¹ x
- tan⁻¹(-x) = -tan⁻¹ x
- cosec⁻¹(-x) = -cosec⁻¹ x
- sec⁻¹(-x) = π - sec⁻¹ x
- cot⁻¹(-x) = π - cot⁻¹ x
🔹 Reciprocal Relations
- sin⁻¹(1/x) = cosec⁻¹ x
- cos⁻¹(1/x) = sec⁻¹ x
- tan⁻¹(1/x) = cot⁻¹ x, x > 0
- cosec⁻¹(1/x) = sin⁻¹ x
- sec⁻¹(1/x) = cos⁻¹ x
- cot⁻¹(1/x) = tan⁻¹ x, x > 0
🔹 Complementary Relations
- sin⁻¹ x + cos⁻¹ x = π/2
- tan⁻¹ x + cot⁻¹ x = π/2
- sec⁻¹ x + cosec⁻¹ x = π/2
🔹 Sum and Difference Formulas
- tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x+y)/(1-xy)], xy < 1
- tan⁻¹ x - tan⁻¹ y = tan⁻¹[(x-y)/(1+xy)], xy > -1
- 2tan⁻¹ x = tan⁻¹(2x/(1-x²)), |x| < 1
- sin⁻¹ x + sin⁻¹ y = sin⁻¹[x√(1-y²) + y√(1-x²)]
- cos⁻¹ x + cos⁻¹ y = cos⁻¹[xy - √(1-x²)√(1-y²)]
🔄 SUBSTITUTIONS FOR SIMPLEST FORM
🔹 Common Substitutions:
| Expression | Substitution |
|---|---|
| √(a² - x²) | x = a sin θ or x = a cos θ |
| √(a² + x²) | x = a tan θ or x = a cot θ |
| √(x² - a²) | x = a sec θ or x = a cosec θ |
| (a - x)/(a + x) or (a + x)/(a - x) | x = a cos 2θ |
| x² + a² or x² - a² | x = a tan θ or x = a sec θ |
💡 IMPORTANT POINTS TO REMEMBER
🔹 While Solving Equations:
- Sometimes, some values of x found may not satisfy the given equation.
- Do not cancel common factors from both sides of an equation.
🔹 Principal Values:
- sin⁻¹ x has principal value in [-π/2, π/2]
- cos⁻¹ x has principal value in [0, π]
- tan⁻¹ x has principal value in (-π/2, π/2)
- cot⁻¹ x has principal value in (0, π)
- sec⁻¹ x has principal value in [0, π] - {π/2}
- cosec⁻¹ x has principal value in [-π/2, π/2] - {0}
📌 QUICK REFERENCE SUMMARY
| Function | Domain | Range | sin⁻¹(sin θ) = θ for θ in |
|---|---|---|---|
| sin⁻¹ x | [-1, 1] | [-π/2, π/2] | [-π/2, π/2] |
| cos⁻¹ x | [-1, 1] | [0, π] | [0, π] |
| tan⁻¹ x | R | (-π/2, π/2) | (-π/2, π/2) |
| cot⁻¹ x | R | (0, π) | (0, π) |
| sec⁻¹ x | (-∞,-1]∪[1,∞) | [0, π]-{π/2} | [0, π]-{π/2} |
| cosec⁻¹ x | (-∞,-1]∪[1,∞) | [-π/2, π/2]-{0} | [-π/2, π/2]-{0} |
📝 MCQ QUESTIONS
Q1. The principal value of sin⁻¹(1/2) is:
(a) π/6 (b) π/3 (c) 5π/6 (d) π/2
Answer: (a) π/6
Solution: sin(π/6) = 1/2, and π/6 ∈ [-π/2, π/2]
Q2. The principal value of cos⁻¹(1/2) is:
(a) π/6 (b) π/3 (c) 2π/3 (d) 5π/6
Answer: (b) π/3
Solution: cos(π/3) = 1/2, and π/3 ∈ [0, π]
Q3. The principal value of tan⁻¹(1) is:
(a) π/4 (b) 3π/4 (c) 5π/4 (d) π/2
Answer: (a) π/4
Q4. The principal value of cos⁻¹(-1/2) is:
(a) π/3 (b) 2π/3 (c) 4π/3 (d) 5π/3
Answer: (b) 2π/3
Q5. sin⁻¹(sin(3π/4)) is equal to:
(a) 3π/4 (b) π/4 (c) π/2 (d) -π/4
Answer: (b) π/4
Solution: sin(3π/4) = 1/√2, sin⁻¹(1/√2) = π/4
Q6. cos⁻¹(cos(5π/4)) is equal to:
(a) 5π/4 (b) 3π/4 (c) π/4 (d) π/2
Answer: (b) 3π/4
Solution: cos(5π/4) = -1/√2, cos⁻¹(-1/√2) = 3π/4
Q7. The value of sin⁻¹(√3/2) is:
(a) π/3 (b) π/6 (c) 2π/3 (d) 5π/6
Answer: (a) π/3
Q8. The value of tan⁻¹(√3) is:
(a) π/6 (b) π/3 (c) π/2 (d) 2π/3
Answer: (b) π/3
Q9. sin⁻¹ x + cos⁻¹ x is equal to:
(a) π/2 (b) π (c) 0 (d) 2π
Answer: (a) π/2
Q10. tan⁻¹ x + cot⁻¹ x is equal to:
(a) π/2 (b) π (c) 0 (d) 2π
Answer: (a) π/2
Q11. sec⁻¹ x + cosec⁻¹ x is equal to:
(a) π/2 (b) π (c) 0 (d) 2π
Answer: (a) π/2
Q12. The domain of sin⁻¹ x is:
(a) [-1, 1] (b) R (c) [0, π] (d) (-1, 1)
Answer: (a) [-1, 1]
Q13. The range of cos⁻¹ x is:
(a) [-π/2, π/2] (b) [0, π] (c) (-π/2, π/2) (d) (0, π)
Answer: (b) [0, π]
Q14. The value of cos⁻¹(cos(3π/4)) is:
(a) 3π/4 (b) π/4 (c) π/2 (d) 5π/4
Answer: (a) 3π/4
Q15. tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) is equal to:
(a) π (b) π/2 (c) 2π (d) 3π/2
Answer: (a) π
Solution: tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) = π
✅ ANSWER KEY - MCQs
| Q.No. | Answer | Q.No. | Answer |
|---|---|---|---|
| 1 | (a) | 9 | (a) |
| 2 | (b) | 10 | (a) |
| 3 | (a) | 11 | (a) |
| 4 | (b) | 12 | (a) |
| 5 | (b) | 13 | (b) |
| 6 | (b) | 14 | (a) |
| 7 | (a) | 15 | (a) |
| 8 | (b) |
📌 KEY FORMULAS SUMMARY
| Formula | Expression |
|---|---|
| Negative Arguments | sin⁻¹(-x) = -sin⁻¹ x, cos⁻¹(-x) = π - cos⁻¹ x |
| Complementary | sin⁻¹ x + cos⁻¹ x = π/2 |
| Complementary | tan⁻¹ x + cot⁻¹ x = π/2 |
| Complementary | sec⁻¹ x + cosec⁻¹ x = π/2 |
| Sum Formula | tan⁻¹ x + tan⁻¹ y = tan⁻¹((x+y)/(1-xy)), xy < 1 |
| Difference Formula | tan⁻¹ x - tan⁻¹ y = tan⁻¹((x-y)/(1+xy)), xy > -1 |
| Double Angle | 2tan⁻¹ x = tan⁻¹(2x/(1-x²)), |x| < 1 |
| sin(sin⁻¹ x) | x, ∀ x ∈ [-1, 1] |
| cos(cos⁻¹ x) | x, ∀ x ∈ [-1, 1] |
| tan(tan⁻¹ x) | x, ∀ x ∈ R |
📚 CBSE | Class XII Mathematics
Chapter 2: Inverse Trigonometric Functions | Study Notes