Trigonometric Equations

Let $S=\{\theta \in(-2 \pi, 2 \pi): \cos \theta+1=\sqrt{3} \sin \theta\}$.
Then $\sum\limits_{\theta \in \mathrm{S}} \theta$ is equal to :

The sum of all the integral values of $p$ such that the equation $3 \sin ^2 x+12 \cos x-3=p, x \in \mathbb{R}$, has at least one solution, is:

If $S=\left\{\theta \in[-\pi, \pi]: \cos \theta \cos \frac{5 \theta}{2}=\cos 7 \theta \cos \frac{7 \theta}{2}\right\}$, then $n(S)$ is equal to $\_\_\_\_$ .

Let $S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a,\ a \in \mathbb{Z} \}$. Then $n(S)$ is equal to:

The number of elements in the set $\left\{x \in\left[0,180^{\circ}\right]: \tan \left(x+100^{\circ}\right)=\tan \left(x+50^{\circ}\right) \tan x \tan \left(x-50^{\circ}\right)\right\}$ is $\_\_\_\_$ .

Number of solutions of $\sqrt{3} \cos 2 \theta+8 \cos \theta+3 \sqrt{3}=0, \theta \in[-3 \pi, 2 \pi]$ is :

The number of solutions of the equation
$ \cos 2\theta \cos \frac{\theta}{2} + \cos \frac{5\theta}{2} = 2\cos^3 \frac{5\theta}{2} $ in $ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] $ is :

\(\text { The number of solutions of the equation } 2 x+3 \tan x=\pi, x \in[-2 \pi, 2 \pi]-\left\{ \pm \frac{\pi}{2}, \pm \frac{3 \pi}{2}\right\} \text { is: }\)

The number of solutions of the equation
$(4-\sqrt{3}) \sin x-2 \sqrt{3} \cos ^2 x=-\frac{4}{1+\sqrt{3}}, x \in\left[-2 \pi, \frac{5 \pi}{2}\right]$ is

If $\theta \in[-2 \pi, 2 \pi]$, then the number of solutions of $2 \sqrt{2} \cos ^2 \theta+(2-\sqrt{6}) \cos \theta-\sqrt{3}=0$, is equal to:

If $\theta \epsilon\left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right]$, then the number of solutions of $\sqrt{3} \operatorname{cosec}^2 \theta-2(\sqrt{3}-1) \operatorname{cosec} \theta-4=0$, is equal to :

The sum of all values of $\theta \in[0,2 \pi]$ satisfying $2 \sin ^2 \theta=\cos 2 \theta$ and $2 \cos ^2 \theta=3 \sin \theta$ is

Let \(|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]\). Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is :

The number of solutions of \(\sin ^2 x+\left(2+2 x-x^2\right) \sin x-3(x-1)^2=0\), where \(-\pi \leq x \leq \pi\), is ________.

Let \(S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\left(\sin ^6 \theta+\cos ^6 \theta\right)=0\right.\) has real roots \(\}\). If \(\alpha\) and \(\beta\) be the smallest and largest elements o...

If \(2 \sin ^3 x+\sin 2 x \cos x+4 \sin x-4=0\) has exactly 3 solutions in the interval \(\left[0, \frac{\mathrm{n} \pi}{2}\right], \mathrm{n} \in \mathrm{N}\), then the roots of the equation \(x^2+\mathrm{n} x+(\mathrm{n}-3)=0\) belong to ...

If \(\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}\) is the solution of \(4 \cos \theta+5 \sin \theta=1\), then the value of \(\tan \alpha\) 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 \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly 7 solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in \mathbf{N}\), then \(\sum_\limits{k=1}^n \frac{k}{2^k}\) is equal to:

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 :

If m and n respectively are the numbers of positive and negative values of \(\theta\) in the interval \([-\pi,\pi]\) that satisfy the equation \(\cos 2\theta \cos {\theta \over 2} = \cos 3\theta \cos {{9\theta } \over 2}\), then mn is equa...

Let \(\mathrm{S = \{ \theta \in [0,2\pi ):\tan (\pi \cos \theta ) + \tan (\pi \sin \theta ) = 0\}}\). Then \(\sum\limits_{\theta \in S} {{{\sin }^2}\left( {\theta + {\pi \over 4}} \right)}\) is equal to __________.

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

\(\beta=\sum_\limits{x \in S} \tan ^{2}\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^{2}\) is equal to :

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 \(n({S_1}) - n({S_2})\) is equal to ______________.

The number of solutions of the equation \(2\theta - {\cos ^2}\theta + \sqrt 2 = 0\) in R is equal to ___________.

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 \(\sin x = {\cos ^2}x\) in the interval (0, 10) is _________.

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 equations $$x^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x+6 \sin ^{2} \theta=0, \theta \...

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 \(S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}\). Then the number of elements in the set $$\mid A=\{\theta \in S: \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (...

Let for some real numbers \(\alpha\) and \(\beta\), \(a = \alpha - i\beta\). If the system of equations \(4ix + (1 + i)y = 0\) and \(8\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)x + \overline a y = 0\) has more than o...

Let \(S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \pi}{6}\right)=-\frac{8}{\sqrt{3}}\right\}\). Then

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( {\sec \left( {{\pi \over 4} + 2\theta } \right)\cos ec\left( {{\pi \over 4} + 2\theta } ...

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 \([0,2 \pi]\) is \(k \pi\), then \(k\) is equal to __________.

The number of values of x in the interval \(\left( {{\pi \over 4},{{7\pi } \over 4}} \right)\) for which \(14\cos e{c^2}x - 2{\sin ^2}x = 21 - 4{\cos ^2}x\) holds, is ____________.

The number of solutions of \(|\cos x|=\sin x\), such that \(-4 \pi \leq x \leq 4 \pi\) is :

Let \(S = \left\{ {\theta \in [ - \pi ,\pi ] - \left\{ { \pm \,\,{\pi \over 2}} \right\}:\sin \theta \tan \theta + \tan \theta = \sin 2\theta } \right\}\). If \(T = \sum\limits_{\theta \, \in \,S}^{} {\cos 2\theta }\), then T + n(S) is...

The number of solutions of the equation \(\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x\), \(x \in [ - 3\pi ,3\pi ]\) is :

The number of solutions of the equation \({32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4}\) is :

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\(\pi\)]. Then \({{8S} \over \pi }\) is equal to ____________.

If \(\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1\), the number of solutions of the given equation when \(x \in \left[ {0,{\pi \over 2}} \right]\) is __________.

The sum of solutions of the equation \({{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|\), \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right) - \left\{ {{\pi \over 4}, - {\pi \over 4}} \right\}\) is :

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 :

All possible values of \(\theta\) \(\in\) [0, 2\(\pi\)] for which sin 2\(\theta\) + tan 2\(\theta\) > 0 lie in :

The number of solutions of sin7x + cos7x = 1, x\(\in\) [0, 4\(\pi\)] is equal to

If n is the number of solutions of the equation \(2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{\pi \over 4} - x} \right) - 1} \right) = 1,x \in [0,\pi ]\) and S is the sum of all these solutions, then the ordered p...

The number of solutions of the equation \(|\cot x| = \cot x + {1 \over {\sin x}}\) in the interval [ 0, 2\(\pi\) ] is

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 :