Composition — sin(sin⁻¹ x), sin⁻¹(sin x)
Inverse Trigonometric Functions
3
JEE Qs
8%
Hard
75
min
Always prioritize the domain and range restrictions for both the inner and outer functions when dealing with compositions of trigonometric and inverse trigonometric functions.
🧮 Key Formulas
✅ Key Points for JEE
- 1For `sin(sin⁻¹ x)`, the expression is defined as `x` ONLY if `x` is within the domain of `sin⁻¹ x`, i.e., `x ∈ [-1, 1]`. Outside this interval, `sin⁻¹ x` is undefined, and hence the composite function is undefined.
- 2For `sin⁻¹(sin x)`, the output of the inverse sine function must ALWAYS lie within its principal value branch, which is `[-pi/2, pi/2]`. The expression equals `x` only if `x` itself is already in this range.
- 3When evaluating `sin⁻¹(sin x)` for `x` outside `[-pi/2, pi/2]`, use trigonometric identities (e.g., `sin(π - θ) = sin θ`, `sin(2π + θ) = sin θ`, `sin(-θ) = -sin θ`) to find an equivalent angle `y` such that `sin y = sin x` and `y ∈ [-pi/2, pi/2]`. Then `sin⁻¹(sin x) = y`.
- 4The graph of `y = sin⁻¹(sin x)` is a periodic, piecewise linear function with a period of `2π`, bounded between `[-pi/2, pi/2]`. Understanding its graph is crucial for solving equations and inequalities.
⚠️ Common Mistakes
- ✕Blindly assuming `sin(sin⁻¹ x) = x` for any `x` without checking the domain `[-1, 1]`. For `x` outside this domain, `sin(sin⁻¹ x)` is undefined.
- ✕Blindly assuming `sin⁻¹(sin x) = x` for any `x` without considering the principal value branch `[-pi/2, pi/2]`. This is the most common mistake for this topic.
- ✕Incorrectly simplifying `sin⁻¹(sin x)` by misapplying trigonometric identities or failing to bring the argument into the principal value branch.
📝 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 11 Maths Ch 3: Trigonometric Functions
- Class 12 Maths Ch 2: Inverse Trigonometric Functions