Class 12 Matrices | NCERT Maths Chapter 3 Notes & Solutions

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Mathematics — INVERSE TRIGONOMETRIC FUNCTIONS

50 Questions Class 12 MCQs

The principal value of \(\sin^{-1}\left(\frac{1}{2}\right)\) is

a a. \(\frac{\pi}{6}\) b b. \(\frac{5\pi}{6}\) c c. \(-\frac{\pi}{6}\) d d. \(\frac{\pi}{3}\)

The principal value of \(\cos^{-1}\left(\frac{1}{2}\right)\) is

a a. \(\frac{\pi}{6}\) b b. \(\frac{\pi}{3}\) c c. \(\frac{2\pi}{3}\) d d. \(\frac{5\pi}{3}\)

The principal value of \(\tan^{-1}(1)\) is

a a. \(\frac{\pi}{4}\) b b. \(\frac{3\pi}{4}\) c c. \(-\frac{\pi}{4}\) d d. \(\frac{5\pi}{4}\)

The principal value of \(\sin^{-1}(-1)\) is

a a. \(\frac{\pi}{2}\) b b. \(-\frac{\pi}{2}\) c c. \(0\) d d. \(\pi\)

The principal value of \(\cos^{-1}(-1)\) is

a a. \(0\) b b. \(\frac{\pi}{2}\) c c. \(\pi\) d d. \(-\pi\)

The principal value of \(\tan^{-1}(-\sqrt{3})\) is

a a. \(\frac{\pi}{3}\) b b. \(-\frac{\pi}{3}\) c c. \(\frac{2\pi}{3}\) d d. \(-\frac{2\pi}{3}\)

The range of \(\sin^{-1}x\) is

a a. \([0,\pi]\) b b. \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) c c. \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) d d. \([-\pi,\pi]\)

The range of \(\cos^{-1}x\) is

a a. \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) b b. \([0,\pi]\) c c. \((0,\pi)\) d d. \([-\pi,\pi]\)

The range of \(\tan^{-1}x\) is

a a. \([0,\pi]\) b b. \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) c c. \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) d d. \([-\pi,\pi]\)

The domain of \(\sin^{-1}x\) is

a a. \(\mathbb{R}\) b b. \([-1,1]\) c c. \((-\infty,-1]\cup[1,\infty)\) d d. \([0,1]\)

The value of \(\sin^{-1}(\sin\frac{2\pi}{3})\) is

a a. \(\frac{2\pi}{3}\) b b. \(\frac{\pi}{3}\) c c. \(-\frac{\pi}{3}\) d d. \(-\frac{2\pi}{3}\)

The value of \(\cos^{-1}(\cos\frac{4\pi}{3})\) is

a a. \(\frac{4\pi}{3}\) b b. \(-\frac{\pi}{3}\) c c. \(\frac{2\pi}{3}\) d d. \(\frac{\pi}{3}\)

The value of \(\tan^{-1}(\tan\frac{3\pi}{4})\) is

a a. \(\frac{3\pi}{4}\) b b. \(-\frac{\pi}{4}\) c c. \(\frac{\pi}{4}\) d d. \(-\frac{3\pi}{4}\)

The value of \(\sin(\sin^{-1}x)\), where \(x\in[-1,1]\), is

a a. \(1-x^2\) b b. \(-x\) c c. \(x\) d d. \(\sqrt{1-x^2}\)

The value of \(\cos(\cos^{-1}x)\), where \(x\in[-1,1]\), is

a a. \(x\) b b. \(-x\) c c. \(\sqrt{1-x^2}\) d d. \(1-x^2\)

The value of \(\tan(\tan^{-1}x)\), \(x\in\mathbb{R}\), is

a a. \(x^2\) b b. \(-x\) c c. \(x\) d d. \(\frac{1}{x}\)

The value of \(\sin^{-1}x+\cos^{-1}x\), for \(x\in[-1,1]\), is

a a. \(0\) b b. \(\frac{\pi}{2}\) c c. \(\pi\) d d. \(x\)

The value of \(\tan^{-1}1+\tan^{-1}1\) is

a a. \(\frac{\pi}{4}\) b b. \(\frac{\pi}{2}\) c c. \(\pi\) d d. \(0\)

The value of \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\) is

a a. \(\frac{\pi}{3}\) b b. \(-\frac{\pi}{3}\) c c. \(\frac{2\pi}{3}\) d d. \(-\frac{2\pi}{3}\)

The value of \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\) is

a a. \(\frac{\pi}{6}\) b b. \(\frac{5\pi}{6}\) c c. \(-\frac{\pi}{6}\) d d. \(-\frac{5\pi}{6}\)

The value of \(\sin^{-1}\frac{3}{5}+\cos^{-1}\frac{3}{5}\) is

a a. \(\frac{\pi}{4}\) b b. \(\frac{\pi}{2}\) c c. \(\frac{\pi}{3}\) d d. \(\pi\)

If \(\theta=\sin^{-1}\frac{3}{5}\), then \(\cos\theta\) is

a a. \(\frac{3}{5}\) b b. \(\frac{4}{5}\) c c. \(\frac{5}{4}\) d d. \(-\frac{4}{5}\)

If \(\theta=\cos^{-1}\frac{5}{13}\), then \(\sin\theta\) is

a a. \(\frac{12}{13}\) b b. \(-\frac{12}{13}\) c c. \(\frac{5}{13}\) d d. \(\frac{13}{12}\)

If \(\theta=\tan^{-1}\frac{3}{4}\), then \(\sin\theta\) is

a a. \(\frac{3}{5}\) b b. \(\frac{4}{5}\) c c. \(\frac{3}{4}\) d d. \(\frac{4}{3}\)

The value of \(\tan^{-1}\frac{1}{2}+\tan^{-1}\frac{1}{3}\) is

a a. \(\frac{\pi}{4}\) b b. \(\tan^{-1}1\) c c. \(\tan^{-1}\frac{5}{6}\) d d. \(\frac{\pi}{6}\)

The value of \(\tan^{-1}\frac{1}{2}-\tan^{-1}\frac{1}{3}\) is

a a. \(\tan^{-1}\frac{1}{5}\) b b. \(\tan^{-1}\frac{1}{6}\) c c. \(\tan^{-1}\frac{1}{7}\) d d. \(\tan^{-1}1\)

The value of \(\tan^{-1}1+\tan^{-1}2\) is

a a. \(\tan^{-1}3\) b b. \(\frac{\pi}{4}\) c c. \(\frac{3\pi}{4}\) d d. \(\frac{\pi}{2}\)

The value of \(\tan^{-1}2+\tan^{-1}3\) is

a a. \(\tan^{-1}5\) b b. \(\pi-\tan^{-1}7\) c c. \(\tan^{-1}(-7)\) d d. \(\frac{\pi}{4}\)

If \(x=\tan^{-1}\frac{1}{2}\) and \(y=\tan^{-1}\frac{1}{3}\), then \(\tan(x+y)\) is

a a. \(\frac{1}{5}\) b b. \(1\) c c. \(\frac{5}{6}\) d d. \(7\)

The value of \(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{4}{5}\) is

a a. \(\frac{\pi}{2}\) b b. \(\frac{\pi}{4}\) c c. \(\frac{\pi}{3}\) d d. \(\pi\)

The value of \(\cos^{-1}\frac{3}{5}-\cos^{-1}\frac{4}{5}\) is

a a. \(\sin^{-1}\frac{7}{25}\) b b. \(\sin^{-1}\frac{1}{25}\) c c. \(\cos^{-1}\frac{7}{25}\) d d. \(\frac{\pi}{2}\)

If \(0

a a. \(0\) b b. \(\frac{\pi}{4}\) c c. \(\frac{\pi}{2}\) d d. \(\pi\)

If \(x>0\), then \(\tan^{-1}x+\tan^{-1}\frac{1}{x}\) is

a a. \(0\) b b. \(\frac{\pi}{4}\) c c. \(\frac{\pi}{2}\) d d. \(\pi\)

If \(x<0\), then \(\tan^{-1}x+\tan^{-1}\frac{1}{x}\) is

a a. \(\frac{\pi}{2}\) b b. \(-\frac{\pi}{2}\) c c. \(0\) d d. \(\pi\)

The value of \(\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{12}{13}\) is

a a. \(\frac{\pi}{2}\) b b. \(\frac{\pi}{3}\) c c. \(\frac{\pi}{4}\) d d. \(\pi\)

If \(x=\sin^{-1}\frac{3}{5}\), then \(\tan x\) is

a a. \(\frac{3}{4}\) b b. \(\frac{4}{3}\) c c. \(\frac{3}{5}\) d d. \(\frac{5}{4}\)

The value of \(\cos^{-1}\left(\cos\frac{7\pi}{6}\right)\) is

a a. \(\frac{7\pi}{6}\) b b. \(\frac{5\pi}{6}\) c c. \(-\frac{5\pi}{6}\) d d. \(\frac{\pi}{6}\)

The value of \(\sin^{-1}\left(\sin\frac{5\pi}{6}\right)\) is

a a. \(\frac{5\pi}{6}\) b b. \(\frac{\pi}{6}\) c c. \(-\frac{\pi}{6}\) d d. \(-\frac{5\pi}{6}\)

If \(\theta=\tan^{-1}\frac{3}{4}\), then \(\sin2\theta\) is

a a. \(\frac{7}{25}\) b b. \(\frac{24}{25}\) c c. \(\frac{3}{4}\) d d. \(\frac{4}{5}\)

If \(\theta=\tan^{-1}\frac{3}{4}\), then \(\cos2\theta\) is

a a. \(\frac{7}{25}\) b b. \(-\frac{7}{25}\) c c. \(\frac{24}{25}\) d d. \(-\frac{24}{25}\)

The value of \(\tan^{-1}\frac{1}{2}+\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{7}\) is

a a. \(\frac{\pi}{6}\) b b. \(\frac{\pi}{4}\) c c. \(\frac{\pi}{3}\) d d. \(\frac{\pi}{2}\)

If \(A=\tan^{-1}\frac12\) and \(B=\tan^{-1}\frac13\), then \(A+B\) equals

a a. \(\tan^{-1}\frac15\) b b. \(\tan^{-1}1\) c c. \(\tan^{-1}\frac56\) d d. \(\tan^{-1}\frac76\)

If \(A=\sin^{-1}\frac35\) and \(B=\cos^{-1}\frac35\), then \(A+B\) is

a a. \(\frac{\pi}{4}\) b b. \(\frac{\pi}{2}\) c c. \(\frac{\pi}{3}\) d d. \(\pi\)

The value of \(\cos^{-1}\frac35+\cos^{-1}\frac45\) is

a a. \(\frac{\pi}{2}\) b b. \(\frac{\pi}{3}\) c c. \(\frac{\pi}{4}\) d d. \(\pi\)

For \(x\in[-1,1]\), the expression \(\cos(\sin^{-1}x)\) equals

a a. \(\sqrt{1-x^2}\) b b. \(-\sqrt{1-x^2}\) c c. \(1-x^2\) d d. \(x\)

For \(x\in[-1,1]\), the expression \(\sin(\cos^{-1}x)\) equals

a a. \(\sqrt{1-x^2}\) b b. \(-\sqrt{1-x^2}\) c c. \(x\) d d. \(1-x^2\)

If \(x\in[-1,1]\), then \(\tan(\sin^{-1}x)\) is

a a. \(\frac{x}{\sqrt{1-x^2}}\) b b. \(\frac{\sqrt{1-x^2}}{x}\) c c. \(\frac{x}{1-x^2}\) d d. \(\frac{1}{\sqrt{1-x^2}}\)

If \(x\in[-1,1]\), then \(\cot(\cos^{-1}x)\) is

a a. \(\frac{x}{\sqrt{1-x^2}}\) b b. \(\frac{\sqrt{1-x^2}}{x}\) c c. \(\frac{1}{x}\) d d. \(\frac{x}{1-x^2}\)

If \(x>0\), the value of \(\tan^{-1}\frac{x-1}{x+1}\) is equal to

a a. \(\tan^{-1}x-\frac{\pi}{4}\) b b. \(\tan^{-1}x+\frac{\pi}{4}\) c c. \(\frac{\pi}{4}-\tan^{-1}x\) d d. \(\pi-\tan^{-1}x\)

For \(x>1\), the value of \(\tan^{-1}\frac{2x}{1-x^2}\) is

a a. \(2\tan^{-1}x\) b b. \(\pi-2\tan^{-1}x\) c c. \(2\tan^{-1}x-\pi\) d d. \(\frac{\pi}{2}-2\tan^{-1}x\)

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Class 12 Matrices | NCERT Maths Chapter 3 Notes & Solutions

Class 12 Matrices | NCERT Maths Chapter 3 Notes & Solutions — Complete Notes & Solutions · academia-aeternum.com

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Frequently Asked Questions

Inverse trigonometric functions return the principal value of an angle corresponding to a given trigonometric ratio. The six inverse functions are \(\sin^{-1}x\), \(\cos^{-1}x\), \(\tan^{-1}x\), \(\cot^{-1}x\), \(\sec^{-1}x\), and \(\csc^{-1}x\).

Trigonometric functions are periodic and not one-one over their entire domains. Restricting the domain to a principal interval makes them one-one, allowing inverse functions to exist.

The principal value range of \(\sin^{-1}x\) is \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\), while its domain is \([-1,1]\).

\(\sin^{-1}x:\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\), \(\cos^{-1}x:[0,\pi]\), \(\tan^{-1}x:\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\), \(\cot^{-1}x:(0,\pi)\), \(\sec^{-1}x:[0,\pi]-\left\{\dfrac{\pi}{2}\right\}\), and \(\csc^{-1}x:\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]-\{0\}\).

The domain of \(\tan^{-1}x\) is the set of all real numbers \(\mathbb{R}\), and its principal value lies in \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\).

First determine whether \(x\) lies in the principal interval \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\). If not, reduce it to an equivalent angle within the principal interval before applying the inverse sine function.

For every \(x\in[-1,1]\), the identity is \(\sin^{-1}x+\cos^{-1}x=\dfrac{\pi}{2}\). It is one of the most frequently used identities in Board and JEE examinations.

Common mistakes include confusing inverse functions with reciprocal functions, ignoring principal value intervals, using \(\sin^{-1}(\sin x)=x\) for all real \(x\), and forgetting domain restrictions.

They are extensively used in differentiation, integration, limits, trigonometric equations, coordinate geometry and calculus. Questions based on principal values and identities are regularly asked in CBSE Board and competitive examinations.

Memorize the principal value intervals, learn standard inverse values, practice principal value problems from different quadrants, revise important identities, and always check the domain before simplifying inverse trigonometric expressions.

The principal value range of \(\sin^{-1}x\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\).

The principal value range of \(\cos^{-1}x\) is \([0,\pi]\).

The principal value range of \(\tan^{-1}x\) is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).

The domain of both \(\sin^{-1}x\) and \(\cos^{-1}x\) is \([-1,1]\).

The domain of \(\tan^{-1}x\) is the set of all real numbers, \(\mathbb{R}\).

Inverse Trigonometric Functions

MCQ Bank Snapshot

Why Practise These MCQs?

Topic-wise MCQ Breakdown

Must-Know Formulae Before You Start

MCQ Solving Strategy

⚠ Common Traps & Errors

Difficulty Ladder

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Mathematics — INVERSE TRIGONOMETRIC FUNCTIONS

Frequently Asked Questions

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INVERSE TRIGONOMETRIC FUNCTIONS — Learning Resources

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Inverse Trigonometric Functions

MCQ Bank Snapshot

Why Practise These MCQs?

Topic-wise MCQ Breakdown

Must-Know Formulae Before You Start

MCQ Solving Strategy

⚠ Common Traps & Errors

Difficulty Ladder

Continue Your Preparation

Mathematics — INVERSE TRIGONOMETRIC FUNCTIONS

Frequently Asked Questions

Q 01 What are inverse trigonometric functions?

Q 02 Why are domains of trigonometric functions restricted before defining their inverses?

Q 03 What is the principal value range of \(\sin^{-1}x\)?

Q 04 What are the principal value intervals of all inverse trigonometric functions?

Q 05 What is the domain of the inverse tangent function?

Q 06 How do you evaluate expressions like \(\sin^{-1}(\sin x)\)?

Q 07 What is the identity relating inverse sine and inverse cosine?

Q 08 What are the most common mistakes in inverse trigonometric functions?

Q 09 How are inverse trigonometric functions useful in CBSE and JEE Mathematics?

Q 10 How can I score full marks in the Inverse Trigonometric Functions chapter?

Q 11 What is the principal value range of \(\sin^{-1}x\)?

Q 12 What is the principal value range of \(\cos^{-1}x\)?

Q 13 What is the principal value range of \(\tan^{-1}x\)?

Q 14 What is the domain of \(\sin^{-1}x\) and \(\cos^{-1}x\)?

Q 15 What is the domain of \(\tan^{-1}x\)?

Recent Posts

INVERSE TRIGONOMETRIC FUNCTIONS — Learning Resources

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