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## Trigonometric Functions & Equations

### Mathematics

#### Previous Years Questions

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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, $$x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} ... JEE Main 2019 (Online) 12th April Morning Slot The value of sin 10º sin30º sin50º sin70º is :- JEE Main 2019 (Online) 9th April Evening Slot The value of cos210° – cos10°cos50° + cos250° is JEE Main 2019 (Online) 9th April Morning Slot Let S = {$$\theta  \in $$[–2$$\pi $$, 2$$\pi $$] : 2cos2$$\theta $$+ 3sin$$\theta $$= 0}. Then the sum of the el... JEE Main 2019 (Online) 9th April Morning Slot If cos($$\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)
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 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)
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
The number of values of $$x$$ in the interval $$\left[ {0,3\pi } \right]\,$$ satisfying the equation $$2{\sin ^2}x + 5\... AIEEE 2006 If$$0 &lt; x &lt; \pi $$and$$\cos x + \sin x = {1 \over 2},$$then$$\tan x$$is AIEEE 2006 A line makes the same angle$$\theta $$, with each of the$$x$$and$$z$$axis. If the angle$$\beta \,$$, which it m... AIEEE 2004 If$$u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}...
AIEEE 2004
Let $$\alpha ,\,\beta$$ be such that $$\pi &lt; \alpha - \beta &lt; 3\pi$$. If $$sin{\mkern 1mu} \alpha + \sin \be... AIEEE 2004 Which one is not periodic AIEEE 2002 The number of solution of$$\tan \,x + \sec \,x = 2\cos \,x$$in$$\left[ {0,\,2\,\pi } \right]$$is AIEEE 2002 The period of$${\sin ^2}\theta $$is AIEEE 2002 ## Numerical More If the sum of solutions of the system of equations$$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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