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## Trigonometric Functions & Equations

### Mathematics

#### Previous Years Questions

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The number of elements in the set $$S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\... JEE Main 2022 (Online) 29th July Evening Shift Let$$S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\righ...
JEE Main 2022 (Online) 27th July Evening Shift
Let $$S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .$$ Then $$n(s) + \sum\limi... JEE Main 2022 (Online) 26th July Morning Shift$$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\f...
JEE Main 2022 (Online) 25th July Evening Shift
The number of solutions of $$|\cos x|=\sin x$$, such that $$-4 \pi \leq x \leq 4 \pi$$ is :
JEE Main 2022 (Online) 25th July Morning Shift
Let $$S = \left\{ {\theta \in [ - \pi ,\pi ] - \left\{ { \pm \,\,{\pi \over 2}} \right\}:\sin \theta \tan \theta + \t... JEE Main 2022 (Online) 24th June Morning Shift The number of solutions of the equation$$\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \righ...
JEE Main 2022 (Online) 24th June Evening Shift
The value of 2sin (12$$^\circ$$) $$-$$ sin (72$$^\circ$$) is :
JEE Main 2022 (Online) 25th June Evening Shift
$$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$$ is equal to :
JEE Main 2022 (Online) 26th June Evening Shift
The value of $$\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi }... JEE Main 2022 (Online) 27th June Morning Shift$$\alpha = \sin 36^\circ $$is a root of which of the following equation? JEE Main 2022 (Online) 27th June Evening Shift Let for some real numbers$$\alpha$$and$$\beta$$,$$a = \alpha - i\beta $$. If the system of equations$$4ix + (1 + i...
JEE Main 2022 (Online) 27th June Evening Shift
If cot$$\alpha$$ = 1 and sec$$\beta$$ = $$- {5 \over 3}$$, where $$\pi ... JEE Main 2022 (Online) 28th June Evening Shift If n is the number of solutions of the equation$$2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{...
JEE Main 2021 (Online) 1st September Evening Shift
The number of solutions of the equation $${32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4}$$ ...
JEE Main 2021 (Online) 31st August Evening Shift
The distance of the point (1, $$-$$2, 3) from the plane x $$-$$ y + z = 5 measured parallel to a line, whose direction r...
JEE Main 2021 (Online) 27th August Morning Shift
The value of $$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8... JEE Main 2021 (Online) 26th August Evening Shift The sum of solutions of the equation$${{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|$$,$$x \in \left( { - {\...
JEE Main 2021 (Online) 26th August Morning Shift
If $$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equ...
JEE Main 2021 (Online) 27th July Morning Shift
The value of $$\cot {\pi \over {24}}$$ is :
JEE Main 2021 (Online) 25th July Evening Shift
If $$\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$$ are in arithmetic progression a...
JEE Main 2021 (Online) 27th July Evening Shift
The sum of all values of x in [0, 2$$\pi$$], for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to :
JEE Main 2021 (Online) 25th July Morning Shift
If 15sin4$$\alpha$$ + 10cos4$$\alpha$$ = 6, for some $$\alpha$$$$\in$$R, then the value of 27sec6$$\alpha$$ + 8cosec6$$\... JEE Main 2021 (Online) 18th March Evening Shift The number of solutions of the equation x + 2tanx =$${\pi \over 2}$$in the interval [0, 2$$\pi$$] is : JEE Main 2021 (Online) 17th March Evening Shift If for x$$\in\left( {0,{\pi \over 2}} \right)$$, log10sinx + log10cosx =$$-$$1 and log10(sinx + cosx) =$${1 \ov...
JEE Main 2021 (Online) 16th March Morning Shift
The number of roots of the equation, (81)sin2x + (81)cos2x = 30 in the interval [ 0, $$\pi$$ ] is equal to :...
JEE Main 2021 (Online) 16th March Morning Shift
If 0 &lt; x, y &lt; $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to :
JEE Main 2021 (Online) 25th February Evening Shift
All possible values of $$\theta$$ $$\in$$ [0, 2$$\pi$$] for which sin 2$$\theta$$ + tan 2$$\theta$$ &gt; 0 lie in :
JEE Main 2021 (Online) 25th February Morning Slot
If $${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$ satisfies the equation...
JEE Main 2021 (Online) 24th February Morning Slot
If L = sin2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$ and M = cos2$$\left( {{\pi... JEE Main 2020 (Online) 5th September Evening Slot If the equation cos4$$\theta $$+ sin4$$\theta $$+$$\lambda $$= 0 has real solutions for$$\theta $$, then$$\lambd...
JEE Main 2020 (Online) 2nd September Evening Slot
If $$x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta }$$ and $$y = \sum\limits_{n = 0... JEE Main 2020 (Online) 9th January Evening Slot The value of$${\cos ^3}\left( {{\pi \over 8}} \right){\cos}\left( {{3\pi \over 8}} \right)$$+$${\sin ^3}\left( {{...
JEE Main 2020 (Online) 9th January Morning Slot
Let S be the set of all $$\alpha$$ $$\in$$ R such that the equation, cos2x + $$\alpha$$sinx = 2$$\alpha$$– 7 has a ...
JEE Main 2019 (Online) 12th April Evening Slot
If [x] denotes the greatest integer $$\le$$ x, then the system of linear equations [sin $$\theta$$]x + [–cos$$\theta ... JEE Main 2019 (Online) 12th April Evening Slot The equation y = sinx sin (x + 2) – sin2 (x + 1) represents a straight line lying in : JEE Main 2019 (Online) 12th April Morning Slot The number of solutions of the equation 1 + sin4 x = cos23x,$$x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} ...
JEE Main 2019 (Online) 12th April Morning Slot
The value of sin 10º sin30º sin50º sin70º is :-
JEE Main 2019 (Online) 9th April Evening Slot
Let S = {$$\theta$$ $$\in$$ [–2$$\pi$$, 2$$\pi$$] : 2cos2$$\theta$$ + 3sin$$\theta$$ = 0}. Then the sum of the el...
JEE Main 2019 (Online) 9th April Morning Slot
The value of cos210° – cos10°cos50° + cos250° is
JEE Main 2019 (Online) 9th April Morning Slot
If cos($$\alpha$$ + $$\beta$$) = 3/5 ,sin ( $$\alpha$$ - $$\beta$$) = 5/13 and 0 &lt; $$\alpha , \beta$$ &lt; $$\pi ... JEE Main 2019 (Online) 8th April Morning Slot The maximum value of 3cos$$\theta $$+ 5sin$$\left( {\theta - {\pi \over 6}} \right)$$for any real value of$$\thet...
JEE Main 2019 (Online) 12th January Morning Slot
The value of $$\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {... JEE Main 2019 (Online) 10th January Evening Slot The sum of all values of$$\theta  \in \left( {0,{\pi \over 2}} \right)$$satisfying sin2 2$$\theta $$+ cos4 ... JEE Main 2019 (Online) 10th January Morning Slot If&nbsp;&nbsp;0$$ \le $$x &lt;$${\pi \over 2}$$, &nbsp;then the number of values of x for which sin x$$-$$sin 2x +... JEE Main 2019 (Online) 9th January Evening Slot For any$$\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$$, the expression$$3{(\cos \theta - \sin \theta...
JEE Main 2019 (Online) 9th January Morning Slot
The number of solutions of sin3x = cos 2x, in the interval $$\left( {{\pi \over 2},\pi } \right)$$ is :
JEE Main 2018 (Online) 15th April Evening Slot
PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of...
JEE Main 2018 (Offline)
If sum of all the solutions of the equation $$8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi... JEE Main 2018 (Offline) Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground ... JEE Main 2017 (Offline) If$$5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$$, then the value of$$\cos 4x$$is JEE Main 2017 (Offline) Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval : JEE Main 2016 (Online) 10th April Morning Slot If&nbsp; m and M are the minimum and the maximum values of 4 +$${1 \over 2}$$sin2 2x$$-$$2cos4 x, x$$ \in $$R, the... JEE Main 2016 (Online) 9th April Morning Slot The number of x$$ \in $$[0, &nbsp; 2$$\pi $$] for which$$\left| {\sqrt {2{{\sin }^4}x + 18{{\cos }^2}x} - \sqrt {...
JEE Main 2016 (Online) 9th April Morning Slot
If $$0 \le x &lt; 2\pi$$, then the number of real values of $$x$$, which satisfy the equation $$\,\cos x + \cos 2x + \c... JEE Main 2016 (Offline) Let$$fk\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$where$$x \in R$$and$$k \ge \,.$$... JEE Main 2014 (Offline)$$ABCD$$is a trapezium such that$$AB$$and$$CD$$are parallel and$$BC \bot CD.$$If$$\angle ADB = \theta ,\,BC = p\$...
JEE Main 2013 (Offline)
The expression $${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$$ can be written...
JEE Main 2013 (Offline)
In a $$\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}$$ If $$3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos... AIEEE 2012 If$$A = {\sin ^2}x + {\cos ^4}x,$$then for all real$$x$$: AIEEE 2011 Let$$\cos \left( {\alpha + \beta } \right) = {4 \over 5}$$and$$\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \ov...
AIEEE 2010
Let A and B denote the statements A: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$ B: $$\sin \alpha + \sin \beta ... AIEEE 2009 If$$0 &lt; x &lt; \pi $$and$$\cos x + \sin x = {1 \over 2},$$then$$\tan x$$is AIEEE 2006 The number of values of$$x$$in the interval$$\left[ {0,3\pi } \right]\,$$satisfying the equation$$2{\sin ^2}x + 5\...
AIEEE 2006
A line makes the same angle $$\theta$$, with each of the $$x$$ and $$z$$ axis. If the angle $$\beta \,$$, which it m...
AIEEE 2004
Let $$\alpha ,\,\beta$$ be such that $$\pi &lt; \alpha - \beta &lt; 3\pi$$. If $$sin{\mkern 1mu} \alpha + \sin \be... AIEEE 2004 If$$u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}...
AIEEE 2004
Which one is not periodic
AIEEE 2002
The number of solution of $$\tan \,x + \sec \,x = 2\cos \,x$$ in $$\left[ {0,\,2\,\pi } \right]$$ is
AIEEE 2002
The period of $${\sin ^2}\theta$$ is
AIEEE 2002

## Numerical

More
Let $$S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2\right\}$$. Then, the su...
JEE Main 2022 (Online) 29th July Morning Shift
Let $$S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$....
JEE Main 2022 (Online) 28th July Evening Shift
If the sum of solutions of the system of equations $$2 \sin ^{2} \theta-\cos 2 \theta=0$$ and $$2 \cos ^{2} \theta+3 \si... JEE Main 2022 (Online) 26th July Evening Shift Let$${S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\} $$and$${S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^...
JEE Main 2022 (Online) 30th June Morning Shift
The number of values of x in the interval $$\left( {{\pi \over 4},{{7\pi } \over 4}} \right)$$ for which $$14\cos e{c^2... JEE Main 2022 (Online) 25th June Morning Shift If$${\sin ^2}(10^\circ )\sin (20^\circ )\sin (40^\circ )\sin (50^\circ )\sin (70^\circ ) = \alpha - {1 \over {16}}\sin...
JEE Main 2022 (Online) 26th June Morning Shift
The number of solutions of the equation $$\sin x = {\cos ^2}x$$ in the interval (0, 10) is _________.
JEE Main 2022 (Online) 29th June Evening Shift
The number of solutions of the equation $$2\theta - {\cos ^2}\theta + \sqrt 2 = 0$$ in R is equal to ___________....
JEE Main 2022 (Online) 29th June Morning Shift
The number of elements in the set $$S = \{ \theta \in [ - 4\pi ,4\pi ]:3{\cos ^2}2\theta + 6\cos 2\theta - 10{\cos ^2... JEE Main 2022 (Online) 29th June Morning Shift Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A$$ \subseteq $$S : A$$\ne\phi$$and ... JEE Main 2021 (Online) 27th August Evening Shift Let z1 and z2 be two complex numbers such that$$\arg ({z_1} - {z_2}) = {\pi \over 4}$$and z1, z2 satisfy the equation... JEE Main 2021 (Online) 27th August Evening Shift The probability distribution of random variable X is given by : .tg {border-collapse:collapse;border-spacing:0;} .tg td... JEE Main 2021 (Online) 27th August Evening Shift Let S be the sum of all solutions (in radians) of the equation$${\sin ^4}\theta + {\cos ^4}\theta - \sin \theta \cos ...
JEE Main 2021 (Online) 27th August Evening Shift
The number of solutions of the equation $$|\cot x| = \cot x + {1 \over {\sin x}}$$ in the interval [ 0, 2$$\pi$$ ] is...
JEE Main 2021 (Online) 18th March Morning Shift
The number of integral values of 'k' for which the equation $$3\sin x + 4\cos x = k + 1$$ has a solution, k$$\in$$R is _...
JEE Main 2021 (Online) 26th February Morning Shift
If $$\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1$$, the number of solutions of the given equation when $$x \in \left... JEE Main 2021 (Online) 26th February Morning Shift If$${{\sqrt 2 \sin \alpha } \over {\sqrt {1 + \cos 2\alpha } }} = {1 \over 7}$$and$$\sqrt {{{1 - \cos 2\beta } \over ...
JEE Main 2020 (Online) 8th January Evening Slot

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