Mathematics · Class 12 Science
Ch 2Inverse Trigonometric Functions — concept-first.
From Chapter 1, the inverse of a function , written , exists only when is both one‑one (injective) and onto (surjective). The trigonometric functions , , , , , fail one or both conditions over their natural domains and ranges.
Key concepts
Hover a concept to preview it and jump to its most relevant questions.
Inverse Sine Principal Value
The equation has infinitely many solutions. If , then could be , , , and so on. To make a genuine function, we must agree on one answer. That agreed-upon answer is called the principal value.
Most relevant questions
- Find the principal value of the following: $ \sin^{-1}\left(-\frac{1}{2}\right) $Free
- Find the principal value of the following: $ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) $Free
- Find the principal value of the following: $ \operatorname{cosec}^{-1}(2) $Free
- If $ \sin^{-1} x = y $, then (A) $ 0 \le y \le \pi $ (B) $ -\frac{\pi}{2} \le y \le \frac{\pi}{2} $ (C) $ 0 < y < \pi $ (D) $ -\frac{\pi}{2}…Preview
- Find the principal value of $\sin^{-1}\left(\dfrac{1}{\sqrt{2}}\right)$.Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
From Chapter 1, the inverse of a function , written , exists only when is both one‑one (injective) and onto (surjective).
Basic Concepts
16 QIn Class XI the six trigonometric functions were studied as mappings on subsets of , each with a specific domain and range:
+−Worked Examplesi2 questions
+−Exercise 2.1i14 questions
- Q1Find the principal value of the following: $ \sin^{-1}\left(-\frac{1}{2}\right) $Free
- Q2Find the principal value of the following: $ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) $Free
- Q3Find the principal value of the following: $ \operatorname{cosec}^{-1}(2) $Free
- Q4Find the principal value of the following: $ \tan^{-1}(-\sqrt{3}) $Preview
- Q5Find the principal value of the following: $ \cos^{-1}\left(-\frac{1}{2}\right) $Preview
- Q6Find the principal value of the following: $ \tan^{-1}(-1) $Preview
- Q7Find the principal value of the following: $ \sec^{-1} \left( \frac{2}{\sqrt{3}} \right) $Preview
- Q8Find the principal value of the following: $ \cot^{-1} \left( \sqrt{3} \right) $Preview
- Q9Find the principal value of the following: $ \cos^{-1} \left( -\frac{1}{\sqrt{2}} \right) $Preview
- Q10Find the principal value of the following: $ \text{cosec}^{-1} \left( -\sqrt{2} \right) $Preview
- Q11Find the value of the following: $ \tan^{-1}(1) + \cos^{-1} \left( -\frac{1}{2} \right) + \sin^{-1} \left( -\frac{1}{2} \right) $Preview
- Q12Find the value of the following: $ \cos^{-1} \left( \frac{1}{2} \right) + 2 \sin^{-1} \left( \frac{1}{2} \right) $Preview
- Q13If $ \sin^{-1} x = y $, then (A) $ 0 \le y \le \pi $ (B) $ -\frac{\pi}{2} \le y \le \frac{\pi}{2} $ (C) $ 0 < y < \pi $ (D) $ -\frac{\pi}{2}…Preview
- Q14$ \tan^{-1} \sqrt{3} - \sec^{-1}(-2) $ is equal to (A) $ \pi $ (B) $ -\frac{\pi}{3} $ (C) $ \frac{\pi}{3} $ (D) $ \frac{2\pi}{3} $Preview
Properties of Inverse Trigonometric Functions
18 QThis section develops the algebraic relationships that hold among inverse trigonometric functions. Every result presented here is valid only within the principal value branches of the corresponding fu…
+−Worked Examplesi3 questions
- Example 3Show that (i) $\sin^{-1}\left(2x\sqrt{1-x^{2}}\right) = 2\sin^{-1}x$, $-\dfrac{1}{\sqrt{2}} \le x \le \dfrac{1}{\sqrt{2}}$ (ii) $\sin^{-1}\l…Free
- Example 4Express $\tan^{-1}\left(\dfrac{\cos x}{1-\sin x}\right)$, $-\dfrac{3\pi}{2} < x < \dfrac{\pi}{2}$ in the simplest form.Preview
- Example 5Write $\cot^{-1}\left(\dfrac{1}{\sqrt{x^{2}-1}}\right)$, $x > 1$ in the simplest form.Preview
+−Exercise 2.2i15 questions
- Q1Find the principal value of the following: $3\sin^{-1} x = \sin^{-1} (3x - 4x^3)$, $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$Free
- Q2$3\cos^{-1} x = \cos^{-1} (4x^3 - 3x)$, $x \in \left[\frac{1}{2}, 1\right]$ Write the following functions in the simplest form:Free
- Q3Find the principal value of the following: $\tan^{-1} \frac{\sqrt{1+x^2}-1}{x}$, $x \neq 0$Free
- Q4Find the principal value of the following: $\tan^{-1} \left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)$, $0 < x < \pi$Preview
- Q5Find the principal value of the following: $\tan^{-1} \left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$, $-\frac{\pi}{4} < x < \frac{3\p…Preview
- Q6Find the principal value of the following: $\tan^{-1} \frac{x}{\sqrt{a^2-x^2}}$, $|x| < a$Preview
- Q7$\tan^{-1} \left(\frac{3a^2x-x^3}{a^3-3ax^2}\right)$, $a > 0$; $-\frac{a}{\sqrt{3}} < x < \frac{a}{\sqrt{3}}$ Find the values of each of the…Preview
- Q8Find the principal value of the following: $\tan^{-1} \left[2\cos \left(2\sin^{-1} \frac{1}{2}\right)\right]$Preview
- Q9Find the value of the following: $\tan \frac{1}{2}\left[\sin^{-1} \frac{2x}{1+x^2} + \cos^{-1} \frac{1-y^2}{1+y^2}\right]$, where $|x| < 1$,…Preview
- Q10Find the principal value of the following: $\sin^{-1}\left(\sin\frac{2\pi}{3}\right)$Preview
- Q11Find the principal value of the following: $\tan^{-1}\left(\tan\frac{3\pi}{4}\right)$Preview
- Q12Find the principal value of the following: $\tan\left(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2}\right)$Preview
- Q13$\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ is equal to (A) $\frac{7\pi}{6}$ (B) $\frac{5\pi}{6}$ (C) $\frac{\pi}{3}$ (D) $\frac{\pi}{6}$Preview
- Q14$\sin\left(\frac{\pi}{3}-\sin^{-1}\left(-\frac{1}{2}\right)\right)$ is equal to (A) $\frac{1}{2}$ (B) $\frac{1}{3}$ (C) $\frac{1}{4}$ (D) $1…Preview
- Q15$\tan^{-1}\sqrt{3}-\cot^{-1}(-\sqrt{3})$ is equal to (A) $\pi$ (B) $-\frac{\pi}{2}$ (C) $0$ (D) $2\sqrt{3}$ Miscellaneous ExamplesPreview
Miscellaneous
15 Q+−Miscellaneous Examplesi1 question
+−Miscellaneous Exercisei14 questions
- Q1Find the principal value of the following: $\cos^{-1} \left(\cos \frac{13\pi}{6}\right)$Free
- Q2Find the value of $\tan^{-1}\left(\tan \dfrac{7\pi}{6}\right)$.Free
- Q3Prove that $2\sin^{-1} \dfrac{3}{5} = \tan^{-1} \dfrac{24}{7}$.Free
- Q4Find the value of the following: $\sin^{-1} \frac{8}{17} + \sin^{-1} \frac{3}{5} = \tan^{-1} \frac{77}{36}$Preview
- Q5Find the value of the following: $\cos^{-1} \frac{4}{5} + \cos^{-1} \frac{12}{13} = \cos^{-1} \frac{33}{65}$Preview
- Q6Find the value of the following: $\cos^{-1} \frac{12}{13} + \sin^{-1} \frac{3}{5} = \sin^{-1} \frac{56}{65}$Preview
- Q7$\tan^{-1} \frac{63}{16} = \sin^{-1} \frac{5}{13} + \cos^{-1} \frac{3}{5}$ Prove thatPreview
- Q8Prove that: $\tan^{-1} \sqrt{x} = \frac{1}{2} \cos^{-1}\left(\frac{1-x}{1+x}\right)$, $x \in [0, 1]$.Preview
- Q9Find the value of the following: $\cot^{-1} \left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right) = \frac…Preview
- Q10$\tan^{-1} \left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1} x$, $-\frac{1}{\sqrt{…Preview
- Q11Find the value of the following: $2\tan^{-1} (\cos x) = \tan^{-1} (2 \operatorname{cosec} x)$Preview
- Q12Solve the following equation: $\tan^{-1}\left(\frac{1-x}{1+x}\right) = \frac{1}{2}\tan^{-1} x$, $(x > 0)$.Preview
- Q13$\sin (\tan^{-1} x)$, $|x|<1$ is equal to (A) $\frac{x}{\sqrt{1-x^2}}$ (B) $\frac{1}{\sqrt{1-x^2}}$ (C) $\frac{1}{\sqrt{1+x^2}}$ (D) $\frac{…Preview
- Q14$\sin^{-1} (1-x) - 2 \sin^{-1} x = \frac{\pi}{2}$, then $x$ is equal to (A) $0, \frac{1}{2}$ (B) $1, \frac{1}{2}$ (C) $0$ (D) $\frac{1}{2}$Preview
Summary
- Principal value branches: Each inverse trigonometric function has a restricted range (principal value branch) to make it single-valued. For example, has range , has , has , etc.
NCERT Exemplar
Higher-order thinking problems from the NCERT Exemplar.
+−Show 55 questionsHide questions55 questions
- Q1Find the value of $\tan^{-1}\left(\tan\frac{5\pi}{6}\right)+\cos^{-1}\left(\cos\frac{13\pi}{6}\right)$.Free
- Q2Evaluate $\cos\left[\cos^{-1}\left(\frac{-\sqrt3}{2}\right)+\frac{\pi}{6}\right]$.Free
- Q3Prove that $\cot\left(\frac{\pi}{4}-2\cot^{-1}3\right)=7$.Free
- Q4Find the value of $\tan^{-1}\left(\frac{-1}{\sqrt3}\right)+\cot^{-1}\left(\frac{1}{\sqrt3}\right)+\tan^{-1}\left[\sin\left(\frac{-\pi}{2}\ri…Preview
- Q5Find the value of $\tan^{-1}\left(\tan\frac{2\pi}{3}\right)$.Preview
- Q6Show that $2\tan^{-1}(-3)=\frac{-\pi}{2}+\tan^{-1}\left(\frac{-4}{3}\right)$.Preview
- Q7Find the real solutions of the equation $\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^2+x+1}=\frac{\pi}{2}$.Preview
- Q8Find the value of the expression $\sin\left(2\tan^{-1}\frac{1}{3}\right)+\cos\left(\tan^{-1}2\sqrt2\right)$.Preview
- Q9If $2\tan^{-1}(\cos\theta)=\tan^{-1}(2\csc\theta)$, then show that $\theta=\frac{\pi}{4}$.Preview
- Q10Show that $\cos\left(2\tan^{-1}\frac{1}{7}\right)=\sin\left(4\tan^{-1}\frac{1}{3}\right)$.Preview
- Q11Solve the following equation $\cos(\tan^{-1}x)=\sin\left(\cot^{-1}\frac{3}{4}\right)$.Preview
- Q12Prove that $\tan^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=\frac{\pi}{4}+\frac{1}{2}\cos^{-1}x^2$.Preview
- Q13Find the simplified form of $\cos^{-1}\left(\frac{3}{5}\cos x+\frac{4}{5}\sin x\right)$, where $x\in\left[\frac{-3\pi}{4},\frac{\pi}{4}\righ…Preview
- Q14Prove that $\sin^{-1}\frac{8}{17}+\sin^{-1}\frac{3}{5}=\sin^{-1}\frac{77}{85}$.Preview
- Q15Show that $\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}=\tan^{-1}\frac{63}{16}$.Preview
- Q16Prove that $\tan^{-1}\frac{1}{4}+\tan^{-1}\frac{2}{9}=\sin^{-1}\frac{1}{\sqrt5}$.Preview
- Q17Find the value of $4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239}$.Preview
- Q18Show that $\tan\left(\frac{1}{2}\sin^{-1}\frac{3}{4}\right)=\frac{4-\sqrt7}{3}$ and justify why the other value $\frac{4+\sqrt7}{3}$ is igno…Preview
- Q19If $a_1, a_2, a_3, \ldots, a_n$ is an arithmetic progression with common difference $d$, then evaluate the following expression: $\tan\left[…Preview
- Q20Which of the following is the principal value branch of $\cos^{-1}x$? (A) $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$ (B) $(0,\pi)$ (C) $[0,…Preview
- Q21Which of the following is the principal value branch of $\csc^{-1}x$? (A) $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$ (B) $[0,\pi]-\left\{\f…Preview
- Q22If $3\tan^{-1}x+\cot^{-1}x=\pi$, then $x$ equals (A) $0$ (B) $1$ (C) $-1$ (D) $\frac{1}{2}$Preview
- Q23The value of $\sin^{-1}\left(\cos\frac{33\pi}{5}\right)$ is (A) $\frac{3\pi}{5}$ (B) $\frac{-7\pi}{5}$ (C) $\frac{\pi}{10}$ (D) $\frac{-\pi}…Preview
- Q24The domain of the function $\cos^{-1}(2x-1)$ is (A) $[0,1]$ (B) $[-1,1]$ (C) $(-1,1)$ (D) $[0,\pi]$Preview
- Q25The domain of the function defined by $f(x)=\sin^{-1}\sqrt{x-1}$ is (A) $[1,2]$ (B) $[-1,1]$ (C) $[0,1]$ (D) none of thesePreview
- Q26If $\cos\left(\sin^{-1}\frac{2}{5}+\cos^{-1}x\right)=0$, then $x$ is equal to (A) $\frac{1}{5}$ (B) $\frac{2}{5}$ (C) $0$ (D) $1$Preview
- Q27The value of $\sin(2\tan^{-1}(0.75))$ is equal to (A) $0.75$ (B) $1.5$ (C) $0.96$ (D) $\sin 1.5$Preview
- Q28The value of $\cos^{-1}\left(\cos\frac{3\pi}{2}\right)$ is equal to (A) $\frac{\pi}{2}$ (B) $\frac{3\pi}{2}$ (C) $\frac{5\pi}{2}$ (D) $\frac…Preview
- Q29The value of the expression $2\sec^{-1}2+\sin^{-1}\frac{1}{2}$ is (A) $\frac{\pi}{6}$ (B) $\frac{5\pi}{6}$ (C) $\frac{7\pi}{6}$ (D) $1$Preview
- Q30If $\tan^{-1}x+\tan^{-1}y=\frac{4\pi}{5}$, then $\cot^{-1}x+\cot^{-1}y$ equals (A) $\frac{\pi}{5}$ (B) $\frac{2\pi}{5}$ (C) $\frac{3\pi}{5}$…Preview
- Q31If $\sin^{-1}\left(\frac{2a}{1+a^2}\right)+\cos^{-1}\left(\frac{1-a^2}{1+a^2}\right)=\tan^{-1}\left(\frac{2x}{1-x^2}\right)$, where $a, x\in…Preview
- Q32The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ is (A) $\frac{25}{24}$ (B) $\frac{25}{7}$ (C) $\frac{24}{25}$ (D) $\frac{7}{24}$Preview
- Q33The value of the expression $\tan\left(\frac{1}{2}\cos^{-1}\frac{2}{\sqrt5}\right)$ is [Hint: $\tan\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta…Preview
- Q34If $|x|\le1$, then $2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ is equal to (A) $4\tan^{-1}x$ (B) $0$ (C) $\frac{\pi}{2}$ (D) $\pi$Preview
- Q35If $\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma=3\pi$, then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals (A) $…Preview
- Q36The number of real solutions of the equation $\sqrt{1+\cos2x}=\sqrt2\,\cos^{-1}(\cos x)$ in $\left[\frac{\pi}{2},\pi\right]$ is (A) $0$ (B)…Preview
- Q37If $\cos^{-1}x>\sin^{-1}x$, then (A) $\frac{1}{\sqrt2}<x\le1$ (B) $0\le x<\frac{1}{\sqrt2}$ (C) $-1\le x<\frac{1}{\sqrt2}$ (D) $x>0$Preview
- Q38The principal value of $\cos^{-1}\left(-\frac{1}{2}\right)$ is __________.Preview
- Q39The value of $\sin^{-1}\left(\sin\frac{3\pi}{5}\right)$ is __________.Preview
- Q40If $\cos(\tan^{-1}x+\cot^{-1}\sqrt3)=0$, then value of $x$ is __________.Preview
- Q41The set of values of $\sec^{-1}\left(\frac{1}{2}\right)$ is __________.Preview
- Q42The principal value of $\tan^{-1}\sqrt3$ is __________.Preview
- Q43The value of $\cos^{-1}\left(\cos\frac{14\pi}{3}\right)$ is __________.Preview
- Q44The value of $\cos(\sin^{-1}x+\cos^{-1}x)$, $|x|\le1$ is __________.Preview
- Q45The value of expression $\tan\left(\frac{\sin^{-1}x+\cos^{-1}x}{2}\right)$, when $x=\frac{\sqrt3}{2}$ is __________.Preview
- Q46If $y=2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ for all $x$, then ____ $<y<$ ____.Preview
- Q47The result $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right)$ is true when value of $xy$ is __________.Preview
- Q48The value of $\cot^{-1}(-x)$ for all $x\in R$ in terms of $\cot^{-1}x$ is __________.Preview
- Q49State True or False: All trigonometric functions have inverse over their respective domains.Preview
- Q50State True or False: The value of the expression $(\cos^{-1}x)^2$ is equal to $\sec^2 x$.Preview
- Q51State True or False: The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in…Preview
- Q52State True or False: The least numerical value, either positive or negative of angle $\theta$ is called principal value of the inverse trigo…Preview
- Q53State True or False: The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric functio…Preview
- Q54State True or False: The minimum value of $n$ for which $\tan^{-1}\frac{n}{\pi}>\frac{\pi}{4}$, $n\in N$, is valid is 5.Preview
- Q55State True or False: The principal value of $\sin^{-1}\left(\cos\left(\sin^{-1}\frac{1}{2}\right)\right)$ is $\frac{\pi}{3}$.Preview
CBSE Sample Papers
Questions from official CBSE sample papers.