Domain & Range — All 6 inverse trig functions
Inverse Trigonometric Functions
3
JEE Qs
8%
Hard
75
min
Master the precise principal value ranges of all six inverse trig functions, as this is fundamental for accurately solving all related problems and simplification questions.
🧮 Key Formulas
✅ Key Points for JEE
- 1Inverse trigonometric functions are defined by restricting the domain of the original trigonometric function to make it bijective (one-to-one and onto), thus ensuring a unique inverse.
- 2The range of an inverse trigonometric function is its Principal Value Branch (PVB) and must be memorized precisely, as it dictates the output for any valid input.
- 3The domain of sin⁻¹(x) and cos⁻¹(x) is restricted to [-1, 1] because the range of sin(x) and cos(x) is [-1, 1].
- 4The domain of sec⁻¹(x) and cosec⁻¹(x) is (-inf, -1] U [1, inf) because the range of sec(x) and cosec(x) excludes values between -1 and 1.
- 5Pay special attention to the excluded values in the range of sec⁻¹(x) (pi/2) and cosec⁻¹(x) (0), which correspond to points where the original functions are undefined in their restricted domains.
⚠️ Common Mistakes
- ✕Confusing the domain and range of inverse trigonometric functions with those of the original trigonometric functions, or with their reciprocals.
- ✕Incorrectly recalling or applying the Principal Value Branches, especially for sec⁻¹(x) and cosec⁻¹(x) where specific points are excluded from the range.
- ✕Failing to check if the argument of an inverse trigonometric function (e.g., 'f(x)' in sin⁻¹(f(x))) lies within its valid domain before performing operations or simplifying.
- ✕Incorrectly interpreting 'inverse' notation, e.g., thinking sin⁻¹(x) is 1/sin(x).
📝 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 2: Relations and Functions
- Class 11 Maths Ch 3: Trigonometric Functions
- Class 12 Maths Ch 2: Inverse Trigonometric Functions