Trigonometric Equations · Mathematics · MHT CET
MCQ (Single Correct Answer)
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 common principal solution of the equations $\sin \theta=-\frac{1}{2}$ and $\tan \theta=\frac{1}{\sqrt{3}}$ is
The principal solution of of $(5+3 \sin \theta)(2 \cos \theta+1)=0$ are
If $\frac{1}{6} \sin \theta, \cos \theta, \tan \theta$ are in G.P., then the general solution of $\theta$ is
The number of solutions of $16^{\sin ^2 x}+16^{\cos ^2 x}=10$ in $0 \leqslant x \leqslant 2 \pi$ are
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 $3 \sin 2 \theta=2 \sin 3 \theta$ 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
If $\tan 3 \theta=\cot \theta$, then $\theta=$
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 values of $x$ in the interval $(0,5 \pi)$ satisfying the equation $3 \sin ^2 x-7 \sin x+2=0$
The solution set of the equation $\tan x+\sec x=2 \cos x$, in the interval $[0,2 \pi]$ is
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 $\alpha+\beta=\frac{\pi}{3}$. Then the value of $\frac{b}{a}$ is
If the equation $\cos ^4 \theta+\sin ^4 \theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval
The general solution of $\sin x+\cos x=1$ is
The number of solutions of $\tan x+\sec x=2 \cos x$ in $[0,2 \pi]$ 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
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
The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ 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 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
The number of solutions, of $2^{1+|\cos x|+|\cos x|^2+\ldots \ldots \cdots \cdots}=4$ in $(-\pi, \pi)$, is
In $(0,2 \pi)$, the number of solutions of $\tan \theta+\sec \theta=2 \cos \theta$ are
The general solution of $\sin x-3 \sin 2 x+\sin 3 x=\cos x-3 \cos 2 x+\cos 3 x$ is
The Solution set of the equation $\sin ^2 \theta-\cos \theta=\frac{1}{4}$ in the interval $[0,2 \pi]$ is
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
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 roots of the equation, $(81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30$ in the interval $[0, \pi]$, is equal to
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
If $\theta$ and $\alpha$ are not odd multiples of $\frac{\pi}{2}$ then $\tan \theta=\tan \alpha$ implies principal solution is
The general solution of $2 \sqrt{3} \cos ^2 \theta=\sin \theta$ is
The principal solutions, of the equation $\sqrt{3} \sec x+2=0$, are
The principal solutions of the equation $$\sec x+\tan x=2 \cos x$$ are
The solution of $$\sin x+\sin 5 x=\sin 3 x$$ in $$(0, \pi / 2)$$ are
If $$(1+\sqrt{1+x}) \tan x=1+\sqrt{1-x}$$, then $$\sin 4 x$$ is
The general solution of the equation $$3 \sec ^2 \theta=2 \operatorname{cosec} \theta$$ is
The solution set of $$8 \cos ^2 \theta+14 \cos \theta+5=0$$, in the interval $$[0,2 \pi]$$, is
If general solution of $$\cos ^2 \theta-2 \sin \theta+\frac{1}{4}=0$$ is $$\theta=\frac{\mathrm{n} \pi}{\mathrm{A}}+(-1)^{\mathrm{n}} \frac{\pi}{\mathrm{B}}, \mathrm{n} \in \mathrm{Z}$$, then $$\mathrm{A}+\mathrm{B}$$ has the
If $$\cos x+\cos y-\cos (x+y)=\frac{3}{2}$$, then
If the general solution of the equation $$\frac{\tan 3 x-1}{\tan 3 x+1}=\sqrt{3}$$ is $$x=\frac{\mathrm{n} \pi}{\mathrm{p}}+\frac{7 \pi}{\mathrm{q}}, \mathrm{n}, \mathrm{p}, \mathrm{q}, \in \mathrm{Z}$$, then $$\frac{p}{q}$$ is
The number of possible solutions of $$\sin \theta+\sin 4 \theta+\sin 7 \theta=0, \theta \in(0, \pi)$$ are
The number of solutions of $$\tan x+\sec x=2 \cos x$$ in $$[0,2 \pi]$$ are
The number of solutions in $$[0,2 \pi]$$ of the equation $$16^{\sin ^2 x}+16^{\cos ^2 x}=10$$ is
The principal solutions of $$\cot x=\sqrt{3}$$ are
If $$3 \sin \theta=2 \sin 3 \theta$$ and $$0< \theta<\pi$$, then $$\sin \theta=$$
The number of solutions of cos2$$\theta$$ = sin$$\theta$$ in (0, 2$$\pi$$) are
If $$\theta+\phi=\alpha$$ and $$\tan \theta=k \tan \phi($$ where $$K>1)$$, then the value of $$\sin (\theta-\phi)$$ is
$$2 \sin \left(\theta+\frac{\pi}{3}\right)=\cos \left(\theta-\frac{\pi}{6}\right)$$, then $$\tan \theta=$$
The principal solutions of $$\sqrt{3} \sec x+2=0$$ are
If $$x \in\left(0, \frac{\pi}{2}\right)$$ and $$x$$ satisfies the equation $$\sin x \cos x=\frac{1}{4}$$, then the values of $$x$$ are
If $2 \cos ^2 \theta+3 \cos \theta=2$, then permissible value of $\cos \theta$ is
The principal solutions of $\cot x=\sqrt{3}$ are
$$\tan A+2 \tan 2 A+4 \tan 4 A+8 \cot 8 A=$$
The values of $x$ in $\left(0, \frac{\pi}{2}\right)$ satisfying the equation $\sin x \cos x=\frac{1}{4}$ are ..........
The number of solutions of $\sin ^2 \theta=\frac{1}{2}$ in $[0, \pi]$ is ..........
Which of the following equations has no solution?