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JEE Main
Mathematics
Trigonometric Equations
Previous Years Questions

The number of solutions of the equation $4 \sin ^2 x-4 \cos ^3 x+9-4 \cos x=0 ; x \in[-2 \pi, 2 \pi]$ is :
The sum of the solutions $$x \in \mathbb{R}$$ of the equation $$\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6$$ is
If $$\alpha,-\frac{\pi}{2} If$$2 \tan ^2 \theta-5 \sec \theta=1$$has exactly 7 solutions in the interval$$\left[0, \frac{n \pi}{2}\right]$$, for the least value of$$n \in \m...
The number of elements in the set $$S=\left\{\theta \in[0,2 \pi]: 3 \cos ^{4} \theta-5 \cos ^{2} \theta-2 \sin ^{6} \theta+2=0\right\}$$ is :...
Let $$S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^{2} x}+9^{\tan ^{2} x}=10\right\}$$ and $$\beta=\sum_\limits{x \in S} \tan... The number of elements in the set$$S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$$is : Let$$S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \...
Let $$S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .$$ Then $$n(s) + \sum\limits_{\theta \in S}^{} {\left( ... The number of solutions of$$|\cos x|=\sin x$$, such that$$-4 \pi \leq x \leq 4 \pi$$is : Let for some real numbers$$\alpha$$and$$\beta$$,$$a = \alpha - i\beta $$. If the system of equations$$4ix + (1 + i)y = 0$$and$$8\left( {\cos {...
The number of solutions of the equation $$\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x$$,...
Let $$S = \left\{ {\theta \in [ - \pi ,\pi ] - \left\{ { \pm \,\,{\pi \over 2}} \right\}:\sin \theta \tan \theta + \tan \theta = \sin 2\theta } \r... If n is the number of solutions of the equation$$2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{\pi \over 4} - x} \right) - 1...
The number of solutions of the equation $${32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4}$$ is :
The sum of solutions of the equation $${{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|$$, $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \... The sum of all values of x in [0, 2$$\pi$$], for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to : The number of solutions of sin7x + cos7x = 1, x$$\in$$[0, 4$$\pi$$] is equal to The number of solutions of the equation x + 2tanx =$${\pi \over 2}$$in the interval [0, 2$$\pi$$] is : The number of roots of the equation, (81)sin2x + (81)cos2x = 30 in the interval [ 0,$$\pi$$] is equal to :... All possible values of$$\theta\in$$[0, 2$$\pi$$] for which sin 2$$\theta$$+ tan 2$$\theta$$> 0 lie in : If [x] denotes the greatest integer$$ \le $$x, then the system of linear equations [sin$$\theta $$]x + [–cos$$\theta $$]y = 0, [cot$$\theta $$]x + ... Let S be the set of all$$\alpha  \in $$R such that the equation, cos2x +$$\alpha $$sinx = 2$$\alpha $$– 7 has a solution. Then S is equal to : The number of solutions of the equation 1 + sin4 x = cos23x,$$x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} \right]$$is :... Let S = {$$\theta  \in $$[–2$$\pi $$, 2$$\pi $$] : 2cos2$$\theta $$+ 3sin$$\theta $$= 0}. Then the sum of the elements of S is The sum of all values of$$\theta  \in \left( {0,{\pi \over 2}} \right)$$satisfying sin2 2$$\theta $$+ cos4 2$$\theta $$=$${3 \over 4}$$... If 0$$ \le $$x <$${\pi \over 2}$$, then the number of values of x for which sin x$$-$$sin 2x + sin 3x = 0, is : If sum of all the solutions of the equation$$8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi \over 6} - x} \right) - {1 \...
The number of solutions of sin3x = cos 2x, in the interval $$\left( {{\pi \over 2},\pi } \right)$$ is :
The number of x $$\in$$ [0,   2$$\pi$$ ] for which $$\left| {\sqrt {2{{\sin }^4}x + 18{{\cos }^2}x} - \sqrt {2{{\cos }^4}x + 18{{\sin }^2}x... If$$0 \le x < 2\pi $$, then the number of real values of$$x$$, which satisfy the equation$$\,\cos x + \cos 2x + \cos 3x + \cos 4x = 0$$is: The number of values of$$x$$in the interval$$\left[ {0,3\pi } \right]\,$$satisfying the equation$$2{\sin ^2}x + 5\sin x - 3 = 0$$is The number of solution of$$\tan \,x + \sec \,x = 2\cos \,x$$in$$\left[ {0,\,2\,\pi } \right]$$is ## Numerical If m and n respectively are the numbers of positive and negative values of$$\theta$$in the interval$$[-\pi,\pi]$$that satisfy the equation$$\cos ...
Let $$\mathrm{S = \{ \theta \in [0,2\pi ):\tan (\pi \cos \theta ) + \tan (\pi \sin \theta ) = 0\}}$$. Then $$\sum\limits_{\theta \in S} {{{\sin }^2}... Let$$S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2\right\}$$. Then, the sum of roots of all the equation... Let$$S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$. Then the number of elements i... If the sum of solutions of the system of equations$$2 \sin ^{2} \theta-\cos 2 \theta=0$$and$$2 \cos ^{2} \theta+3 \sin \theta=0$$in the interval ... Let$${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 ^7}x = 1\} $$Then$$n({S_1}) -...
The number of solutions of the equation $$\sin x = {\cos ^2}x$$ in the interval (0, 10) is _________.
The number of elements in the set $$S = \{ \theta \in [ - 4\pi ,4\pi ]:3{\cos ^2}2\theta + 6\cos 2\theta - 10{\cos ^2}\theta + 5 = 0\}$$ is _____...
The number of solutions of the equation $$2\theta - {\cos ^2}\theta + \sqrt 2 = 0$$ in R is equal to ___________.
The number of values of x in the interval $$\left( {{\pi \over 4},{{7\pi } \over 4}} \right)$$ for which $$14\cos e{c^2}x - 2{\sin ^2}x = 21 - 4{\cos... Let S be the sum of all solutions (in radians) of the equation$${\sin ^4}\theta + {\cos ^4}\theta - \sin \theta \cos \theta = 0$$in [0, 4$$\pi$$]... The number of solutions of the equation$$|\cot x| = \cot x + {1 \over {\sin x}}$$in the interval [ 0, 2$$\pi$$] is If$$\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1$$, the number of solutions of the given equation when$$x \in \left[ {0,{\pi \over 2}} \right]...
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