Trigonometric Equations

The number of common solutions of the pair of equations $2\sin^2\theta - 2\cos^2\theta + 1 = 0$ and $2\sin^2\theta + 3\sin\theta - 2 = 0$ in the interval $[0, 2\pi]$, is

The general solution of $\cos x + \sin x = \cos 2x + \sin 2x$ is $x = np\pi$ or $x = \dfrac{nq\pi}{3} + \dfrac{\pi}{6}$ for $n \in Z$ then $p : q = $

Let $x \in [0, 6\pi]$ satisfy the equation $\cos x - \sin x = -1$. If $x = k\left(\dfrac{\pi}{3}\right)$ where $k \in N$, then find the number of possible values of $k$ is .....

If the equation $\sin 3\theta - \cos^2\theta = \dfrac{1}{4}$ and $\theta \in [0, \pi]$, then the number of solutions is...

If $0 \leq x \leq \dfrac{\pi}{2}$ , then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

The general solution of the equation $\cot\theta \cdot \cot 2\theta = 1$ is...

If $\cos(p\theta) + \cos(q\theta) = 0$ and $p \neq q$, then the general value of $\theta$ (where n is any integer) is...

If $\sec 4\theta - \sec 2\theta = 2$, then $\theta =$

The number of solutions in $\left[0, \dfrac{\pi}{2}\right)$ of the equation $\cos 3x \cdot \tan 5x = \sin 7x$ is

The general solution of $\cos\theta - \sin\theta = 1$ is

If $\sin 3\alpha = 4\sin\alpha \cdot \sin(x+\alpha) \cdot \sin(x-\alpha)$ where $\alpha \neq n\pi, n \in Z$, then all possible values of $x$ are given as

The principal solutions of $\text{cosec}(\theta) = -2$ are...

The general solution of $\cos 4\theta = \cos 3\theta$ is $\theta = $.....

The general solutions of the equation $\tan ^2 \theta+\sec 2 \theta=1$ are

If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then $\sin \left(\frac{\pi}{4}+\theta\right)=$

The principal solution of of $(5+3 \sin \theta)(2 \cos \theta+1)=0$ are

The number of values of $x$ in the interval $[0,3 \pi]$ satisfying the equation $2 \sin ^2 x+5 \sin x-3=0$ is

The common principal solution of the equations $\sin \theta=-\frac{1}{2}$ and $\tan \theta=\frac{1}{\sqrt{3}}$ is

The number of values of $x$ in the interval $[0,3 \pi]$ satisfying the equation $2 \sin ^2 x+5 \sin x-3=0$ is

The number of solutions of $16^{\sin ^2 x}+16^{\cos ^2 x}=10$ in $0 \leqslant x \leqslant 2 \pi$ are

If $\frac{1}{6} \sin \theta, \cos \theta, \tan \theta$ are in G.P., then the general solution of $\theta$ is

If $\sin \left(\frac{\pi}{4} \cot \theta\right)=\cos \left(\frac{\pi}{4} \tan \theta\right)$, then the general solution of $\theta$ is

If $\tan 3 \theta=\cot \theta$, then $\theta=$

If $3 \sin 2 \theta=2 \sin 3 \theta$ and $0<\theta<\pi$, then the value of $\sin \theta$ is equal to

The possible values of $\theta \in(0, \pi)$ such that $\sin \theta+\sin (4 \theta)+\sin (7 \theta)=0$ are

The general solution of $\sin x-3 \sin 2 x+\sin 3 x=\cos x-3 \cos 2 x+\cos 3 x$ is

In $(0,2 \pi)$, the number of solutions of $\tan \theta+\sec \theta=2 \cos \theta$ are

The number of solutions, of $2^{1+|\cos x|+|\cos x|^2+\ldots \ldots \cdots \cdots}=4$ in $(-\pi, \pi)$, is

The number of integral values of $k$, for which the equation $7 \cos x+5 \sin x=2 \mathrm{k}+1$ has a solution, is

The smallest positive value of $x$ in degrees satisfying the
equation $\tan \left(x+100^{\circ}\right)=\tan \left(x+50^{\circ}\right) \tan (x) \tan \left(x-50^{\circ}\right)$ is

Let $2 \sin ^2 x+3 \sin x-2>0$ and $x^2-x-2<0$ ($x$ is measured in radians).
Then $x$ lies in the interval

The Solution set of the equation $\sin ^2 \theta-\cos \theta=\frac{1}{4}$ in the interval $[0,2 \pi]$ is

If $\theta$ and $\alpha$ are not odd multiples of $\frac{\pi}{2}$ then $\tan \theta=\tan \alpha$ implies principal solution is

Let $2 \sin ^2 x+3 \sin x-2>0$ and $x^2-x-2<0$. ( $x$ is measured in radians). The $x$ lies in the interval

The number of roots of the equation, $(81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30$ in the interval $[0, \pi]$, is equal to

Let $S=\left\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set S is equal to

The number of all values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfying the equation $(1-\tan \theta)(1+\tan \theta) \sec ^2 \theta+2 \tan ^2 \theta=0$ is

The principal solutions, of the equation $\sqrt{3} \sec x+2=0$, are

The general solution of $2 \sqrt{3} \cos ^2 \theta=\sin \theta$ is

The number of values of $x$ in the interval $(0,5 \pi)$ satisfying the equation $3 \sin ^2 x-7 \sin x+2=0$

The general solution of the equation $\sqrt{3} \cos \theta+\sin \theta=\sqrt{2}$ is

Let $P=\{\theta / \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}$ and $Q=\{\theta / \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}$ be two sets, then

The number of integral values of k for which the equation $7\cos x+5\sin x=2k+1$ has a solution, is

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} \mathrm{a} \cos x+2 b \sin x=c$, $x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{...

If the equation $\cos ^4 \theta+\sin ^4 \theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval

The solution set of the equation $\tan x+\sec x=2 \cos x$, in the interval $[0,2 \pi]$ is

If angle $\theta$ in $[0,2 \pi]$ satisfies both the equations $\cot \theta=\sqrt{3}$ and $\sqrt{3} \sec \theta+2=0$, then $\theta$ is equal to

The number of solutions of $\tan x+\sec x=2 \cos x$ in $[0,2 \pi]$ is

The general solution of $\sin x+\cos x=1$ is

If for certain $x, 3 \cos x \neq 2 \sin x$, then the general solution of, $\sin ^2 x-\cos 2 x=2-\sin 2 x$, is