Properties — sin⁻¹ + cos⁻¹ = π/2, etc.
Inverse Trigonometric Functions
3
JEE Qs
8%
Hard
45
min
Always check the domain of the argument 'x' for each inverse trigonometric function before applying these properties to avoid common errors.
🧮 Key Formulas
✅ Key Points for JEE
- 1These identities are valid only for the principal value branch of the respective inverse trigonometric functions and within their specified domains.
- 2Always verify that the argument 'x' falls within the valid domain for each function before applying these properties.
- 3These properties are crucial for simplifying expressions involving inverse trigonometric functions, especially in calculus (differentiation, integration) and solving equations.
- 4Understanding the underlying proof (e.g., let sin⁻¹x = θ, then x = sinθ = cos(π/2 - θ)) helps in recalling the properties and their domain restrictions.
⚠️ Common Mistakes
- ✕Ignoring the domain restrictions for 'x', leading to incorrect application of the identity.
- ✕Confusing these sum properties with other identities, such as those involving sums like tan⁻¹x + tan⁻¹y.
- ✕Assuming the properties hold true for arguments outside the principal value range without proper justification or conversion.
📝 Practice Questions
See allQ20.If α > β > γ > 0, then the expression cot−1 {β (α−β) } + cot−1 {γ (β−γ) } + cot−1 {α (γ−α) } equal to : (1) π (2) 0 (3) π 2 −(α + β + γ) (4) 3π L.
Q21.Let S = {x : cos−1 x = π + sin−1 x + sin−1(2x + 1)}. Then ∑x∈ S(2x −1)2 is equal to ______.
Q11.Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum is : values of 16 ((sec−1 x)2 + (cosec−1 x)2) (1) 24π2 (2) 22π2 (3) 31π2 (4) 18π2
Q20.If π 2 ≤x ≤3π4 , then cos−1 ( 1213 cos x + 135 sin x) is equal to (1) x −tan−1 43 (2) x + tan−1 45 (3) x −tan−1 125 (4) x + tan−1 125
Q10. cos (sin−1 35 + sin−1 135 + sin−1 3365 ) is equal to: (1) 1 (2) 0 (3) 32 (4) 33 65 65
Q86.Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2 sin−1 x + 3 cos−1 x = 2π5 , is _______
NCERT Chapters
- Class 12 Maths Ch 2: Inverse Trigonometric Functions