Class 12 Maths · Chapter 2
Unit: Relations and Functions
Tentative Study Share: 4.0 Marks
Chapter 2: Inverse Trigonometric Functions — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Inverse Trigonometric Functions.
NCERT Exercises in Chapter 2: Inverse Trigonometric Functions
📘 NCERT Concept Notes — Inverse Trigonometric Functions
01 · An inverse needs a chosen branch
Trigonometric functions repeat values. Restricting to a principal branch makes each inverse single-valued. Always compare your final angle with the correct range.
02 · Undoing is conditional
sin(sin⁻¹x)=x holds on the inverse domain. In the other order, sin⁻¹(sin θ)=θ requires θ to be in the principal range. Use symmetry or periodicity first when it is outside.
⚡ Key Formulas & Results — Inverse Trigonometric Functions
Sine inverse
Recall the domain and principal range of sine inverse.
\(\sin^{-1}:[-1,1]\to[-\pi/2,\pi/2]\)
Cosine inverse
Recall the domain and principal range of cosine inverse.
\(\cos^{-1}:[-1,1]\to[0,\pi]\)
Tangent inverse
Recall the domain and principal range of tangent inverse.
\(\tan^{-1}:\mathbb R\to(-\pi/2,\pi/2)\)
Cotangent inverse
Recall the domain and principal range of cotangent inverse.
\(\cot^{-1}:\mathbb R\to(0,\pi)\)
Secant inverse
Recall the domain and principal range of secant inverse.
\(\sec^{-1}:\mathbb R\setminus(-1,1)\to[0,\pi]\setminus\{\pi/2\}\)
Cosecant inverse
Recall the domain and principal range of cosecant inverse.
\(\operatorname{cosec}^{-1}:\mathbb R\setminus(-1,1)\to[-\pi/2,\pi/2]\setminus\{0\}\)
Two different compositions
When does the inverse actually return the original angle?
\(\sin(\sin^{-1}x)=x,\quad\sin^{-1}(\sin\theta)=\theta\)
Triple sine identity
When can you bring the factor 3 outside sine inverse?
\(\sin^{-1}(3x-4x^3)=3\sin^{-1}x\)
Triple cosine identity
What domain makes the triple cosine identity a principal-value identity?
\(\cos^{-1}(4x^3-3x)=3\cos^{-1}x\)
A useful arctangent simplification
Which substitution simplifies (√(1+x²)−1)/x?
\(\tan^{-1}\!\frac{\sqrt{1+x^2}-1}{x}=\frac12\tan^{-1}x\)
🎯 CBSE 2026 Board PYQs from Inverse Trigonometric Functions
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q1
If 2 cos⁻¹ x = y, then
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q25
Simplify: tan⁻¹( (cos2x − sin2x)/(cos2x + sin2x) ), 0 < x < π/4. OR Evaluate: tan( sin⁻¹1 − cos⁻¹(−1/2)…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q12
If tan⁻¹x=3y, which interval contains y?
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q23
Simplify: tan⁻¹( (cos2x − sin2x)/(cos2x + sin2x) ), 0 < x < π/4. OR Evaluate: tan( sin⁻¹1 − cos⁻¹(−1/2)…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q25
Simplify \(\cot^{-1}\sqrt{(1+\cos 2x)/(1-\cos 2x)}\) for \(0<x<\pi/2\). OR Evaluate…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q5
If y=sin⁻¹x+π, find the range of y.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q21
Simplify: tan⁻¹( (cos2x − sin2x)/(cos2x + sin2x) ), 0 < x < π/4. OR Evaluate: tan( sin⁻¹1 − cos⁻¹(−1/2)…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q23
Simplify \(\sin^{-1}\sqrt{(1+\cos 2x)/2}\), for \(0<x<\pi/2\). OR Evaluate…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q34
Find the domain of g(x)=arccos(x²−1), solve g(x)=π/3, and give the range of a nonprincipal inverse…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q32
For p(x)=arcsin(1−2x²), find its domain, solve p(x)=π/6 and give the range of 2p(x)+π/2.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q33
Find the domain of q(x)=arccos(4x²−3), solve q(x)=0, and find the range of 3q(x)−π.
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q1
Find the domain of arccos(2x−5).
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q23
Evaluate \(\sin[\cot^{-1}(\sqrt2\cos(\tan^{-1}1))]\). OR On {1,2,3}, is R={(1,1),(3,3),(1,2)}…
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q12
Find the domain of arcsin(1−2x).
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q21
Evaluate \(\sin[\cot^{-1}(\sqrt2\cos(\tan^{-1}1))]\). OR On {1,2,3}, is R={(1,1),(3,3),(1,2)}…
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q5
Find the domain of arccos(4x+1).
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q24
Evaluate \(\sin[\cot^{-1}(\sqrt2\cos(\tan^{-1}1))]\). OR On {1,2,3}, is R={(1,1),(3,3),(1,2)}…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q1
Which inverse trigonometric function is shown?
📝 Chapterwise Practice Questions — Inverse Trigonometric Functions (105 Questions)
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}(\cos(13\pi/5))\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Find the full real solution set of \(\sin^{-1}x>\cos^{-1}x\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin[\pi/2-\sin^{-1}(-1/2)]\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For \(x\in[-1,1]\setminus\{0\}\), simplify \(\cot(\sin^{-1}x)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}(\cos(\pi/9))\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\theta=\sin^{-1}(\sin(-600^\circ))\), giving the answer in radians.
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(2\sec^{-1}2+\sin^{-1}(1/2)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\cot[\cos^{-1}(7/25)]\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}[\cos(33\pi/5)]\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\cot[\tfrac12\sin^{-1}(\sqrt3/2)]\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For real x, simplify \(\sin(\cot^{-1}x)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
If \(y=\tan^{-1}x\), x real, which interval contains precisely its possible values?
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For \(y=\cot^{-1}x\) with x<0, which interval is exactly the range of y?
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For |x|<1, simplify \(\sin(\tan^{-1}x)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
If \(\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma=3\pi\), find…
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For \(\pi<x<3\pi/2\), simplify \(\tan^{-1}\frac{\sqrt{1+\cos x}+\sqrt{1-\cos x}}{\sqrt{1+\cos…
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
How many real solutions of \(\sqrt{1+\cos2x}=\sqrt2\,\cos^{-1}(\cos x)\) lie in \([\pi/2,\pi]\)?
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For 0≤x≤1, simplify \(\cos^{-1}(2x^2-1)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
If \(\cos[\sin^{-1}(3/5)+\cos^{-1}x]=0\), find x.
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Which is the principal value branch of \(\operatorname{cosec}^{-1}x\)?
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Find the real domain of \(\cos^{-1}(2x-1)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Find the real domain of \(f(x)=\sin^{-1}\sqrt{x-1}\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin[\pi/3+\sin^{-1}(1/2)]\).
Class 12 Inverse Trigonometric Functions · Assertion Reason
Assertion: The real domain of f(x)=sin⁻¹x+cos x is [−1,1]. Reason: The domain of a function is the set…