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 principal value of $\tan^{-1}\Big(\tan\frac{3\pi}{5}\Big)$ is:
- A
$\frac{2\pi}{5}$
- ✓
$\frac{-2\pi}{5}$
- C
$\frac{3\pi}{5}$
- D
$\frac{-3\pi}{5}$
Answer: B.
View full solution →$\sin\Big\{2\cos^{-1}\Big(\frac{-3}{5}\Big)\Big\}$ is equal to:
- A
$\frac{6}{25}$
- B
$\frac{24}{25}$
- C
$\frac{4}{5}$
- ✓
$-\frac{24}{25}$
Answer: D.
View full solution →$\tan^{-1}(\sqrt{3})$
- A
$\frac{\pi}{6}$
- ✓
$\frac{\pi}{3}$
- C
$\frac{2\pi}{3}$
- D
$\frac{5\pi}{6}$
Answer: B.
View full solution →If $\cos^{-1}\text{x}>\sin^{-1}\text{x},$ then:
- ✓
$\frac{1}{\sqrt2}<\text{x}\leq1$
- B
$0\leq\text{x}\leq\frac{1}{\sqrt2}$
- C
$-1\leq\text{x}<\frac{1}{\sqrt2}$
- D
$\text{x}>0$
Answer: A.
View full solution →Choose the correct answer from the given four options.The value of the expression $\tan\Big(\frac{1}{2}\cos^{-1}\frac{2}{\sqrt{5}}\Big)$ is:
Answer: B.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\big(\frac{2}{5}\big)+\tan^{-1}\big(\frac{3}{7}\big)=\frac{\pi}{4}.$
Reason: $\tan^{-1}\big(\frac{\text{x}}{\text{y}}\big)+\tan^{-1}\big(\frac{\text{y}-\text{x}}{\text{y}+\text{x}}\big)=\frac{\pi}{4}.$
- ✓
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.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $\text{x}=\frac{1}{5\sqrt{2}}$ then $\{\text{x}\cos(\cot^{-1}\text{x})+\sin(\cot^{-1}\text{x})\}^{2}=\frac{51}{50}.$
Reason: $\tan\Big[\cos^{-1}\Big(\frac{1}{5\sqrt{2}}\Big)-\sin^{-1}\Big(\frac{4}{\sqrt{17}}\Big)\Big]=\frac{29}{3}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- ✓
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.
Answer: B.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\big(\frac{3}{4}\big)+\tan^{-1}\big(\frac{1}{7}\big)=\frac{\pi}{4}.$
Reason: For $x > 0, y > 0, xy < 1, \tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big).$
- ✓
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.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $0<\text{x}\leq\frac{\pi}{2},$ then $\sin^{-1}(\cos\text{x})+\cos^{-1}(\sin\text{x})=\pi-2\text{x}.$
Reason: $\cos^{-1}\text{x}=\frac{\pi}{2}-\sin^{-1}\text{x} $ for all $\text{x}\in[-1,1].$
- ✓
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.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\big[\text{x}+\sqrt{1+\text{x}^{2}}\big]=\frac{\pi}{2}-\frac{1}{2}\cot^{-1}.$
Reason: $\sin^{2}\Big[2\tan^{-1}\sqrt{\frac{1+\text{x}}{1-\text{x}}}\Big]=1-\text{x}^{2}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- ✓
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.
Answer: B.
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 →