Sample QuestionsInverse Trigonometric Functions questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The value of $\sin ^{-1}\left(\cos \frac{13 \pi}{5}\right)$ is
- A
$-\frac{3 \pi}{5}$
- B
$-\frac{\pi}{10}$
- C
$\frac{3 \pi}{5}$
- D
$\frac{\pi}{10}$
View full solution →$\sin \left[\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right]$ is equal to
- A
$\frac{1}{2}$
- B
$\frac{1}{3}$
- C
- D
View full solution →$\sin \left(\tan ^{-1} x\right)$, where $|x|<1$, 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
$\frac{x}{\sqrt{1+x^2}}$
View full solution →Simplest form of
$\tan ^{-1}\left(\frac{\sqrt{1+\cos x}+\sqrt{1-\cos x}}{\sqrt{1+\cos x}-\sqrt{1-\cos x}}\right)$,$\pi$< x<$\frac{3 \pi}{2}$ is
View full solution →If $\tan ^{-1} x=y$, then
- A
- B
$\frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$
- C
$\frac{-\pi}{2}$ < y < $\frac{\pi}{2}$
- D
$y \in\left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}$
View full solution →Assertion (A): Domain of $y=\cos ^{-1}(x)$ is $[-1,1]$.
Reason $(R)$ : The range of the principal value branch of $y=\cos ^{-1}(x)$ is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$.
- A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true, but Reason (R) is false.
- D
Assertion (A) is false, but Reason (R) is true.
View full solution →Assertion $(A)$ : The range of the function $f(x)=2 \sin ^{-1} x+\frac{3 \pi}{2}$, where $x \in[-1,1]$, is $\left[\frac{\pi}{2}, \frac{5 \pi}{2}\right]$.
Reason $(R)$ : The range of the principal value branch of $\sin ^{-1}(x)$ is $[0, \pi]$
View full solution →Assertion (A): The principal value of $\cot ^{-1}(\sqrt{3})$ is $\frac{\pi}{6}$.
Reason $( R )$ : Domain of $\cot ^{-1} x$ is $R -\{-1,1\}$.
- A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true and Reason (R) is false.
- D
Assertion (A) is false and Reason (R) is true.
View full solution →Assertion (A) : All trigonometric functions have their inverses over their respective domains.
Reason $( R )$ : The inverse of $\tan ^{-1} x$ exists for some $x \in R$.
- A
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- B
Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
- C
Assertion (A) is true but Reason (R) is false.
- D
Assertion (A) is false but Reason (R) is true.
View full solution →Assertion (A): The domain of the function $\sec ^{-1} 2 x$ is \[\left(-\infty,-\frac{1}{2}\right] \cup\left[\frac{1}{2}, \infty\right) .\] Reason $(R): \sec ^{-1}(-2)=-\frac{\pi}{4}$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A$.
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
View full solution →Prove that: ${\cot ^{ - 1}}\left( {\frac{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} }}{{\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right) = \frac{x}{2}$, $x \in \left( {0,\frac{\pi }{4}} \right)$
View full solution →Prove that: $\tan ^ { - 1 } \sqrt { x } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 1 - x } { 1 + x } \right) , x \in ( 0,1 ).$
View full solution →Prove that: ${\tan ^{ - 1}}\frac{{63}}{{16}} = {\sin ^{ - 1}}\frac{5}{{13}} + {\cos ^{ - 1}}\frac{3}{5}$
View full solution →Prove that $\cos ^ { - 1 } \left( \frac { 12 } { 13 } \right) + \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right) = \sin ^ { - 1 } \left( \frac { 56 } { 65 } \right).$
View full solution →Prove that $ \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) + \cos ^ { - 1 } \left( \frac { 12 } { 13 } \right) = \cos ^ { - 1 } \left( \frac { 33 } { 65 } \right).$
View full solution →Find the value of ${\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\left( {\frac{1}{2}} \right)} \right)} \right]$.
View full solution →Write the function in the simplest form: ${\tan ^{ - 1}}\frac{x}{{\sqrt {{a^2} - {x^2}} }},\left| x \right| < a$
View full solution →Write the function in the simplest form: $\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),-\frac{\pi}{4}<x<\frac{3\pi}{4}$
View full solution →Write the function in the simplest form: ${\tan ^{ - 1}}\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} ,\;0<x < \pi $
View full solution →Write the function in the simplest form: ${\tan ^{ - 1}}\frac{{\sqrt {1 + {x^2}} - 1}}{x},x \ne 0$.
View full solution →Find the value of $\tan \frac{1}{2}\left[ {{{\sin }^{ - 1}}\frac{{2x}}{{1 + {x^2}}} + {{\cos }^{ - 1}}\frac{{1 - {y^2}}}{{1 + {y^2}}}} \right]$, |x| < 1, y > 0 and xy < 1
View full solution →Write the function in the simplest form: ${\tan ^{ - 1}}\left( {\frac{{3{a^2}x - {x^3}}}{{{a^3} - 3a{x^2}}}} \right),\;a > 0,\left( { - \frac{a}{{\sqrt 3 }} < x < \frac{a}{{\sqrt 3 }}} \right)$
View full solution →