Class 12 • mathematics • Chapter 2
INVERSE TRIGONOMETRIC FUNCTIONS
True & False Quiz
Restrict the domain. Reclaim the inverse.
✓True
✗False
25
Questions
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Ch.2
Chapter
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XII
Class
Why True & False for INVERSE TRIGONOMETRIC FUNCTIONS?
How this format sharpens your conceptual clarity
🔵 Inverse trig functions exist only after restricting the domain so the original function becomes bijective on that branch.
✅ T/F questions target principal value ranges and identities like sin⁻¹x + cos⁻¹x = π/2 — frequently misquoted.
🎯 Trap: tan⁻¹x is defined for ALL real x (range (−π/2, π/2)), unlike sin⁻¹ and cos⁻¹ which need x∈[−1,1].
📋
Read each statement carefully. Click True or False — instant feedback with explanation appears. Submit anytime; unattempted questions are marked Skipped.
Q 1
\(\sin^{-1}x\) is defined for every real number \(x\).
Q 2
The principal value range of \(\sin^{-1}x\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\).
Q 3
The principal value range of \(\cos^{-1}x\) is \([0,\pi]\).
Q 4
The principal value range of \(\tan^{-1}x\) includes both \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\).
Q 5
\(\sin^{-1}(-x)=-\sin^{-1}x\) for \(x\in[-1,1]\).
Q 6
\(\cos^{-1}(-x)=-\cos^{-1}x\) for \(x\in[-1,1]\).
Q 7
\(\tan^{-1}(-x)=-\tan^{-1}x\) for every \(x\in\mathbb{R}\).
Q 8
\(\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\) for \(x\in[-1,1]\).
Q 9
\(\sin(\sin^{-1}x)=x\) for every real number \(x\).
Q 10
\(\cos(\cos^{-1}x)=x\) for \(x\in[-1,1]\).
Q 11
\(\tan(\tan^{-1}x)=x\) for every \(x\in\mathbb{R}\).
Q 12
\(\sin^{-1}(\sin\theta)=\theta\) for every real \(\theta\).
Q 13
\(\cos^{-1}(\cos\theta)=\theta\) for every \(\theta\in[0,\pi]\).
Q 14
\(\tan^{-1}(\tan\theta)=\theta\) whenever \(\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).
Q 15
\(\sin^{-1}\frac{1}{2}=\frac{5\pi}{6}\).
Q 16
\(\cos^{-1}\left(-\frac{1}{2}\right)=\frac{2\pi}{3}\).
Q 17
\(\tan^{-1}1=\frac{5\pi}{4}\).
Q 18
For \(x>0\), \(\tan^{-1}x+\tan^{-1}\frac{1}{x}=\frac{\pi}{2}\).
Q 19
For \(x<0\), \(\tan^{-1}x+\tan^{-1}\frac{1}{x}=\frac{\pi}{2}\).
Q 20
\(\tan^{-1}a+\tan^{-1}b=\tan^{-1}\left(\frac{a+b}{1-ab}\right)\) for all real \(a,b\).
Q 21
\(\tan^{-1}\frac12+\tan^{-1}\frac13=\frac{\pi}{4}\).
Q 22
\(\cos(\sin^{-1}x)=\sqrt{1-x^2}\) for \(x\in[-1,1]\).
Q 23
\(\sin(\cos^{-1}x)=-\sqrt{1-x^2}\) for \(x\in[-1,1]\).
Q 24
If \(\theta=\sin^{-1}\frac35\), then \(\tan\theta=\frac43\).
Q 25
For \(x>1\), \(\tan^{-1}\left(\frac{2x}{1-x^2}\right)=2\tan^{-1}x-\pi\).
Key Takeaways — INVERSE TRIGONOMETRIC FUNCTIONS
Core facts for CBSE Boards & JEE
1
Principal value range of sin⁻¹x is [−π/2, π/2]; of cos⁻¹x is [0, π].
2
sin⁻¹x + cos⁻¹x = π/2 for all x ∈ [−1,1].
3
tan⁻¹x + cot⁻¹x = π/2 for all real x; range of tan⁻¹x is (−π/2, π/2).
4
sin⁻¹(sin x) = x holds only when x ∈ [−π/2, π/2], not for all real x.
5
tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)] holds only when xy < 1.
6
sec⁻¹x is defined for |x| ≥ 1; its range excludes π/2 to keep the function well-defined.