# Inverse Trigonometric Functions · Mathematics · JEE Main

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## Numerical

JEE Main 2024 (Online) 9th April Evening Shift
Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $$2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \p... JEE Main 2024 (Online) 6th April Morning Shift For$$n \in \mathrm{N}$$, if$$\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^{-1} n=\frac{\pi}{4}$$, then$$n$$is equal to ________. JEE Main 2023 (Online) 13th April Evening Shift For$$x \in(-1,1]$$, the number of solutions of the equation$$\sin ^{-1} x=2 \tan ^{-1} x$$is equal to __________. JEE Main 2023 (Online) 13th April Morning Shift If$$S=\left\{x \in \mathbb{R}: \sin ^{-1}\left(\frac{x+1}{\sqrt{x^{2}+2 x+2}}\right)-\sin ^{-1}\left(\frac{x}{\sqrt{x^{2}+1}}\right)=\frac{\pi}{4}\ri...
JEE Main 2023 (Online) 10th April Evening Shift
If the domain of the function $$f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$$ is $$[\alpha, \beta) \mathrm{U}(\gamma, \delta]$$, then $$|3 \alpha+10... JEE Main 2023 (Online) 25th January Morning Shift If the sum of all the solutions of$${\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right) + {\cot ^{ - 1}}\left( {{{1 - {x^2}} \over {2x}}} \right)...
JEE Main 2022 (Online) 27th July Morning Shift
For $$k \in \mathbb{R}$$, let the solutions of the equation $$\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\r... JEE Main 2022 (Online) 25th July Evening Shift Let$$x = \sin (2{\tan ^{ - 1}}\alpha )$$and$$y = \sin \left( {{1 \over 2}{{\tan }^{ - 1}}{4 \over 3}} \right)$$. If$$S = \{ a \in R:{y^2} = 1 - x\...
JEE Main 2022 (Online) 29th June Morning Shift
$$50\tan \left( {3{{\tan }^{ - 1}}\left( {{1 \over 2}} \right) + 2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right) + 4\sqrt 2 \tan \left... ## MCQ (Single Correct Answer) JEE Main 2024 (Online) 4th April Evening Shift Given that the inverse trigonometric function assumes principal values only. Let$$x, y$$be any two real numbers in$$[-1,1]$$such that$$\cos ^{-1}...
JEE Main 2024 (Online) 4th April Morning Shift
If the domain of the function $$\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$$ is $$(\al... JEE Main 2024 (Online) 31st January Evening Shift If$$a=\sin ^{-1}(\sin (5))$$and$$b=\cos ^{-1}(\cos (5))$$, then$$a^2+b^2$$is equal to JEE Main 2024 (Online) 31st January Morning Shift For$$\alpha, \beta, \gamma \neq 0$$, if$$\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$$and$$(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)...
JEE Main 2024 (Online) 29th January Evening Shift
Let $$x=\frac{m}{n}$$ ($$m, n$$ are co-prime natural numbers) be a solution of the equation $$\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$$ and let $... JEE Main 2024 (Online) 27th January Evening Shift Considering only the principal values of inverse trigonometric functions, the number of positive real values of $$x$$ satisfying $$\tan ^{-1}(x)+\tan ... JEE Main 2023 (Online) 15th April Morning Shift If the domain of the function f(x)=\log _{e}\left(4 x^{2}+11 x+6\right)+\sin ^{-1}(4 x+3)+\cos ^{-1}\left(\frac{10 x+6}{3}\right) is (\alpha, \bet... JEE Main 2023 (Online) 1st February Evening Shift Let$$S = \left\{ {x \in R:0 If $$\mathrm{n(S)}$$ denotes the number of elements in $$\mathrm{S}$$ then :... JEE Main 2023 (Online) 1st February Morning Shift Let $$S$$ be the set of all solutions of the equation $$\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}... JEE Main 2023 (Online) 31st January Evening Shift Let (a, b) \subset(0,2 \pi) be the largest interval for which \sin ^{-1}(\sin \theta)-\cos ^{-1}(\sin \theta)>0, \theta \in(0,2 \pi), holds. If ... JEE Main 2023 (Online) 31st January Morning Shift If$${\sin ^{ - 1}}{\alpha \over {17}} + {\cos ^{ - 1}}{4 \over 5} - {\tan ^{ - 1}}{{77} \over {36}} = 0,0 ... JEE Main 2023 (Online) 24th January Morning Shift $${\tan ^{ - 1}}\left( {{{1 + \sqrt 3 } \over {3 + \sqrt 3 }}} \right) + {\sec ^{ - 1}}\left( {\sqrt {{{8 + 4\sqrt 3 } \over {6 + 3\sqrt 3 }}} } \righ... JEE Main 2022 (Online) 29th July Evening Shift The domain of the function$$f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$$is : JEE Main 2022 (Online) 28th July Evening Shift The sum of the absolute maximum and absolute minimum values of the function$$f(x)=\tan ^{-1}(\sin x-\cos x)$$in the interval$$[0, \pi]$$is :... JEE Main 2022 (Online) 28th July Morning Shift Considering only the principal values of the inverse trigonometric functions, the domain of the function$$f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{... JEE Main 2022 (Online) 28th July Morning Shift Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $$\cos ^{-1}(x)-2 \sin ^{-1}(x)=... JEE Main 2022 (Online) 27th July Evening Shift The domain of the function$$f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$$, where [t] is t... JEE Main 2022 (Online) 26th July Evening Shift If$$0 ... JEE Main 2022 (Online) 26th July Morning Shift $$\tan \left(2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{\sqrt{5}}{2}+2 \tan ^{-1} \frac{1}{8}\right)$$ is equal to : JEE Main 2022 (Online) 30th June Morning Shift Let m and M respectively be the minimum and the maximum values of $$f(x) = {\sin ^{ - 1}}2x + \sin 2x + {\cos ^{ - 1}}2x + \cos 2x,\,x \in \left[ {0,{... JEE Main 2022 (Online) 30th June Morning Shift Let$$\alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right)$$and$$\beta ... JEE Main 2022 (Online) 29th June Morning Shift The domain of the function $${\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)$$ is : JEE Main 2022 (Online) 27th June Evening Shift The value of $$\cot \left( {\sum\limits_{n = 1}^{50} {{{\tan }^{ - 1}}\left( {{1 \over {1 + n + {n^2}}}} \right)} } \right)$$ is : JEE Main 2022 (Online) 27th June Morning Shift $${\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over... JEE Main 2022 (Online) 26th June Evening Shift If the inverse trigonometric functions take principal values then$${\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over ... JEE Main 2022 (Online) 25th June Evening Shift The value of $${\tan ^{ - 1}}\left( {{{\cos \left( {{{15\pi } \over 4}} \right) - 1} \over {\sin \left( {{\pi \over 4}} \right)}}} \right)$$ is equal... JEE Main 2022 (Online) 24th June Evening Shift Let $$x * y = {x^2} + {y^3}$$ and $$(x * 1) * 1 = x * (1 * 1)$$. Then a value of $$2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + ... JEE Main 2022 (Online) 24th June Morning Shift The set of all values of k for which$${({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R$$, is the interval : JEE Main 2022 (Online) 24th June Morning Shift The domain of the function$$f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}$$is... JEE Main 2021 (Online) 1st September Evening Shift$${\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))$$is equal to :(The inverse trigonometric functions take the prin... JEE Main 2021 (Online) 31st August Evening Shift The domain of the function$$f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x +... JEE Main 2021 (Online) 27th August Evening Shift Let M and m respectively be the maximum and minimum values of the function f(x) = tan$$-$$1 (sin x + cos x) in $$\left[ {0,{\pi \over 2}} \right]$$, ... JEE Main 2021 (Online) 27th August Morning Shift If $${({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2} = a$$; 0 < x < 1, a $$\ne$$ 0, then the value of 2x2 $$-$$ 1 is : JEE Main 2021 (Online) 26th August Evening Shift The domain of the function $${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$$ is : JEE Main 2021 (Online) 26th August Evening Shift If $$\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p}$$, then the value of tan p is : JEE Main 2021 (Online) 22th July Evening Shift If the domain of the function $$f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}... JEE Main 2021 (Online) 20th July Evening Shift The value of$$\tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \right)} \right)$$is equal to : JEE Main 2021 (Online) 20th July Morning Shift The number of real roots of the equation$${\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4}$$is : JEE Main 2021 (Online) 17th March Evening Shift The number of solutions of the equation$${\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x... JEE Main 2021 (Online) 17th March Morning Shift The sum of possible values of x for tan$$-$$1(x + 1) + cot$$-$$1$$\left( {{1 \over {x - 1}}} \right)$$ = tan$$-$$1$$\left( {{8 \over {31}}} \right)$$ ... JEE Main 2021 (Online) 17th March Morning Shift If cot$$-$$1($$\alpha$$) = cot$$-$$1 2 + cot$$-$$1 8 + cot$$-$$1 18 + cot$$-$$1 32 + ...... upto 100 terms, then $$\alpha$$ is :... JEE Main 2021 (Online) 16th March Evening Shift Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy $${\sin ^{ - 1}}\left( {... JEE Main 2021 (Online) 26th February Evening Shift If 0 < a, b < 1, and tan$$-$$1a + tan$$-$$1b =$${\pi \over 4}$$, then the value of$$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \l... JEE Main 2021 (Online) 26th February Morning Shift If $${{{{\sin }^1}x} \over a} = {{{{\cos }^{ - 1}}x} \over b} = {{{{\tan }^{ - 1}}y} \over c}$$; $$0 < x < 1$$, then the value of $$\cos \left( ... JEE Main 2021 (Online) 25th February Evening Shift cosec$$\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]$$is equal to : JEE Main 2021 (Online) 24th February Evening Shift A possible value of$$\tan \left( {{1 \over 4}{{\sin }^{ - 1}}{{\sqrt {63} } \over 8}} \right)$$is : JEE Main 2020 (Online) 5th September Morning Slot If S is the sum of the first 10 terms of the series$${\tan ^{ - 1}}\left( {{1 \over 3}} \right) + {\tan ^{ - 1}}\left( {{1 \over 7}} \right) + {\tan... JEE Main 2020 (Online) 3rd September Morning Slot 2$$\pi$$ - $$\left( {{{\sin }^{ - 1}}{4 \over 5} + {{\sin }^{ - 1}}{5 \over {13}} + {{\sin }^{ - 1}}{{16} \over {65}}} \right)$$ is equal to : JEE Main 2020 (Online) 2nd September Morning Slot The domain of the function f(x) = $${\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)$$ is (– $$\infty$$, -a]$$\cup$$[a, $$... JEE Main 2019 (Online) 12th April Morning Slot The value of$${\sin ^{ - 1}}\left( {{{12} \over {13}}} \right) - {\sin ^{ - 1}}\left( {{3 \over 5}} \right)$$is equal to : JEE Main 2019 (Online) 10th April Evening Slot If$${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha $$,where –1$$ \le $$x$$ \le $$1, – 2$$ \le $$y$$ \le $$2, x$$ \le {y \over 2}$...
JEE Main 2019 (Online) 8th April Morning Slot
If $$\alpha = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)$$, $$\beta = {\tan ^{ - 1}}\left( {{1 \over 3}} \right)$$ where $$0 < \alpha ,\beta <... JEE Main 2019 (Online) 12th January Morning Slot Considering only the principal values of inverse functions, the set A = { x$$ \ge $$0: tan$$-$$1(2x) + tan$$-$$1(3x) =$${\pi \over 4}$$}... JEE Main 2019 (Online) 11th January Evening Slot All x satisfying the inequality (cot–1 x)2– 7(cot–1 x) + 10 > 0, lie in the interval : JEE Main 2019 (Online) 10th January Evening Slot The value of$$\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} \right)$$is : JEE Main 2019 (Online) 9th January Evening Slot If x = sin$$-$$1(sin10) and y = cos$$-$$1(cos10), then y$$-$$x is equal to : JEE Main 2019 (Online) 9th January Morning Slot If$${\cos ^{ - 1}}\left( {{2 \over {3x}}} \right) + {\cos ^{ - 1}}\left( {{3 \over {4x}}} \right) = {\pi \over 2}$$(x >$$3 \over 4$$), then x i... JEE Main 2017 (Online) 9th April Morning Slot A value of x satisfying the equation sin[cot−1 (1+ x)] = cos [tan−1 x], is : JEE Main 2017 (Online) 8th April Morning Slot The value of tan-1$$\left[ {{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} } \over {\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right],\left| x \righ...
JEE Main 2015 (Offline)
Let $${\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right),$$ where $$\left| x \right| < {1 \over {\sqrt 3... JEE Main 2013 (Offline) If$$x, y, z$$are in A.P. and$${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$$and$${\tan ^{ - 1}}z$$are also in A.P., then : AIEEE 2008 The value of$$cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)$$is : AIEEE 2007 If sin-1$$\left( {{x \over 5}} \right)$$+ cosec-1$$\left( {{5 \over 4}} \right)$$=$${\pi \over 2}$$, then the value of x is :... AIEEE 2005 If$${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,$$then$$4{x^2} - 4xy\cos \alpha + {y^2}$$is equal to : AIEEE 2003 The trigonometric equation$${\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a$$has a solution for : AIEEE 2002$${\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x, then sin x is equal to :

## MCQ (More than One Correct Answer)

JEE Main 2023 (Online) 30th January Evening Shift
Let $a_{1}=1, a_{2}, a_{3}, a_{4}, \ldots .$. be consecutive natural numbers. Then \$\tan ^{-1}\left(\frac{1}{1+a_{1} a_{2}}\right)+\tan ^{-1}\left(\f...
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