Sample QuestionsINVERSE TRIGNOMETRIC 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:
- $\frac{2\pi}{5}$
- $\frac{-2\pi}{5}$
- $\frac{3\pi}{5}$
- $\frac{-3\pi}{5}$
View full solution →If $\cos^{-1}\frac{\text{x}}{3}+\cos^{-1}\frac{\text{y}}{2}=\frac{\theta}{2},$ then, $4\text{x}^2-12\text{xy}\cos^2\frac{\theta}{2}+9\text{y}^2=$
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$36$
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$36-36\cos\theta$
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$18-18\cos\theta$
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$18+18\cos\theta$
View full solution →The equation $\sin^{-1}\text{x}-\cos^{-1}\text{x}=\cos^{-1}(\frac{\sqrt3}{2})$ has:
- Nique solution.
- No solution.
- Infinitely many solution.
- None of these.
View full solution →The value of $\cos^{-1}\Big(\cos\frac{5\pi}{3}\Big)+\sin^{-1}\Big(\sin\frac{5\pi}{3}\Big)$ is:
- $\frac{\pi}{2}$
- $\frac{5\pi}{3}$
- $\frac{10\pi}{3}$
- $0$
View full solution →Choose the correct answer from the given four options.
If $|\text{x}|\leq1,$ then $2\tan^{-1}\text{x}+\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ is equal to:
- $4\tan^{-1}\text{x}$
- $0$
- $\frac{\pi}{2}$
- $\pi$
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: $\sin^{-1}\frac{8}{17}+\sin^{-1}\frac{3}{5}=\sin^{-1}\frac{77}{85}.$
Reason: $\sin^{-1}\text{x}+\sin^{-1}\text{y}=\sin^{-1}\big(\text{x}\sqrt{1-\text{y}^{2}}+\text{y}\sqrt{1-\text{x}^{2}}\big)$ for $-1\leq\text{x},\text{y}\leq1,\text{x}^{2}+\text{y}^{2}\leq1.$
- 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.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
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: $\cos^{-1}\text{x}-\sin^{-1}\text{x}=0,$ then $\text{x}=\frac{1}{\sqrt{2}}.$
Reason: $\cot^{-1}\text{x}+\sin^{-1}\text{x}=\frac{\pi}{2}.$
- 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.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
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.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
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: $\tan\Big[\cos^{-1}\Big(\frac{1}{5\sqrt{2}}\Big)-\sin^{-1}\Big(\frac{4}{\sqrt{17}}\Big)\Big]=\frac{29}{3}.$
- 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.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
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.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
View full solution →If $\tan^{-1}\text{x} +\tan^{-1}\text{y} = \frac{\pi}{4},\text{xy} < 1,$then write the value of x + y + xy.
View full solution →Write the value of $\tan^{-1}\bigg[2\sin\bigg(2\cos^{-1}\frac{\sqrt{3}}{2}\bigg)\bigg].$
View full solution →Write the principal value of $\tan^{-1}(\sqrt{3}) - \cot^{-1}( - \sqrt{3}).$
View full solution →Find the principal value of $\tan^{–1} \sqrt{3} – sec^{–1}(– 2).$
View full solution →What is the principal value of $\cot^{-1 }\Bigg(\cos\frac{2\pi}{3}\Bigg)+\sin^{-1}\Bigg(\sin\frac{2\pi}{3}\Bigg)?$
View full solution →Prove that:
$3\sin^{-1}\text{x}=\sin^{-1}(3\text{x}-4\text{x}^3),\text{x}\in\Big[-\frac{1}{2},\frac{1}{2}\Big]$
View full solution →Find the principal values:
$\sec^{-1}\bigg(\frac{2}{\sqrt{3}}\bigg)$
View full solution →Write the value of $\cos\big(\sin^{-1}\text{x}+\cos^{-1}\text{x}\big),|\text{x}|\leq1$
View full solution →Evaluate the following:
$\cot^{-1}\frac{1}{\sqrt3}-\text{cosec}^{-1}(-2)+\sec^{-1}\Big(\frac{2}{\sqrt3}\Big)$
View full solution →Write the value of $\tan^{-1}\sqrt3+\cot^{-1}\sqrt3$
View full solution →Evaluate:
$\tan\Bigg\{ 2\tan ^{-1} \bigg(\frac{1}{5}\bigg) + \frac{\pi}{4}\Bigg\}$
View full solution →If $\tan^{-1} \frac{\text{x - 3}}{\text{x - 4}} + \tan^{-1} \frac{\text{x + 3}}{\text{x + 4}} = \frac{\pi}{4},$ then find the value of x.
View full solution →Solve the equation for $x: \sin^{-1}x + \sin^{-1}(1 - x) = \cos^{-1}x$
View full solution →Solve the following equation:$\cos(\tan^{-1}\text{x}) = \sin\bigg(\cot^{-1}\frac{3}{4}\bigg).$
View full solution →Show that: $\tan\bigg(\frac{1}{2}\sin^{-1}\frac{3}{4}\bigg) = \frac{4-\sqrt{7}}{3}.$
View full solution →Prove the following results:
$\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}=\tan^{-1}\frac{63}{16}$
View full solution →Show that $2\tan^{-1}\text{x}+\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}$ is constant for $\text{x}\geq1,$ find that constant.
View full solution →Prove the following results:
$2\sin^{-1}\frac{3}{5}-\tan^{-1}\frac{17}{31}=\frac{\pi}{4}$
View full solution →Solve the following equation for x:
$\cot^{-1}\text{x}-\cot^{-1}(\text{x}+2)=\frac{\pi}{12},\text{x}>0$
View full solution →Prove the following results:
$\sin^{-1}\frac{12}{13}+\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{63}{16}=\pi$
View full solution →Fill in the blank.If $\text{y}=2\tan^{-1}\text{x}+\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big),$ then ____ < y < ____.
View full solution →Fill in the blank.If $\cos\Big(\tan^{-1}\text{x}+\cot^{-1}\sqrt{3}\Big)=0,$ then value of x is __________.
View full solution →Fill in the blank.The principal value of $\cos^{-1}\Big(-\frac{1}{2}\Big)$ is __________.
View full solution →Fill in the blank.
The set of values of $\sec^{-1}\Big(\frac{1}{2}\Big)$ is __________.
View full solution →Fill in the blank.
The principal value of $\tan^{-1}\sqrt{3}$ is __________.
View full solution →State True or False for the statement. The value of the expression $(\cos^{-1}x)^2$ is equal to $\sec^2x.$
View full solution →State True or False for the statement.
The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions.
State True or False for the statement.
The minimum value of n for which $\tan^{-1}\frac{\text{n}}{\pi}>\frac{\pi}{4},\ \text{n}\in\text{N},$ is valid is 5.
View full solution →State True or False for the statement.
The principal value of $\sin^{-1}\Big[\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big]$ is $\frac{\pi}{3}.$
View full solution →State True or False for the statement.
All trigonometric functions have inverse over their respective domains.