Trigonometric equations practice · Easy

Solution of the system of equations $x+y=\frac{\pi}{4}, \tan x+\tan y=1$ is

The number of real solutions of $2\,\,\sin \left( {{e^x}} \right) = {5^x} + {5^{ - x}}$ in $\left[ {\mathrm{0, 1}} \right]\mathrm{ }\,\mathrm{is /}\,\mathrm{are}$

The general solution of $\tan 3\;x = 1$ is

The number of solutions of ${\sin ^7}x + {\cos ^7}x = 1,\,\,x \in [0,\,4\pi ]$ is equal to

If $\sin \,\,4A - \cos \,\,2A = \cos \,\,4A - \sin \,\,2A,\;\,\left( {0 < A < \frac{\pi }{4}} \right)$, then the value of $\tan \,\,4A$ is

If the solution of the equation $\log _{\cos x} \cot x+4 \log _{\sin x}$ $\tan x=1, x \in\left(0, \frac{\pi}{2}\right)$, is $\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ is equal to

The most general value of $\mathrm{\theta }$ satisfying the equation ${(1 + 2\sin \mathrm{\theta })^2} + \,\,{(\sqrt 3 \tan \mathrm{\theta } - 1)^2} = 0$ are given by

The set of values of $x$ for which $\frac{\tan 3 x-\tan 2 x}{1+\tan 3 x \tan 2 x}=1$ is

The number of values of $x$ for which $\sin \,\,2x + \cos \,\,4x = 2$, is

If $5\,\,\cos \,\,2\theta + 2\,\,{\cos ^2}\,\,\frac{\theta }{2} + 1 = 0$, when $(0 < \theta < \pi )$, then the values of $\theta$ are

Total number of solutions of $[\sin x]=\cos x$, where [ ] denotes the greatest integer function in $[0,3 \pi]$, is

The general solution for $\theta$ if $\sin \left(2 \theta+\frac{\pi}{6}\right)+\cos \left(\theta+\frac{5 \pi}{6}\right)=2$ is

$\log \left(\sin 1^{\circ}\right) \log \left(\sin 2^{\circ}\right) \log \left(\sin 3^{\circ}\right) \ldots \log \left(\sin 179^{\circ}\right)$ is equal to

The number of values of $x \in \left[ {0,\,2\;\pi } \right]$ that satisfy $\cot \,\,x - \mathrm{cosec\;}\,\,x = 2\sin \,\,x$ is

If $-\pi \leq x \leq \pi,-\pi \leq y \leq \pi$ and $\cos x+\cos y=2$, then the value $\cos (x-y)$ is

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

The number of solutions for the equation $\sin \,\,2x + \cos \,\,4x = 2$ is

If $\tan \,\,2x = \tan \,\,\frac{2}{x}$, then the value of $x$ is

If $\sin ^2 \theta+3 \cos \theta-2=0$, then $\cos ^3 \theta+\sec ^3 \theta=\ldots \ldots, \theta \neq(2 k+1) \frac{\pi}{2}, k \in Z$

Total number of integral values of $n$ so that $\sin x(\sin x+\cos x)=n$ has atleast one solution, is

Consider the following statements.

I. Equal numbers may be added to (or subtracted from) both sides of an inequality.

II. When both sides are multiplied (or divided) by a negative number, then the inequality is reversed.

Choose the correct option.

If the equation ${\sin ^2}\,\,\mathrm{\theta } - \cos \,\,\mathrm{\theta } = \frac{1}{4}$, then the value of $\mathrm{\theta }$ lying in the interval $0\,\, \le \,\,\mathrm{\theta }\,\, \le \,\,2\mathrm{\pi }$ is

If $2\,\,{\cos ^2}\,\,x + 3\,\,\sin \,\,x - 3 = 0,\;\,0 \le \,\,x\,\, \le 180^\circ$, then the value of $x$ is

If $\sin \,\,2x\,\,\cos \,\,2x\,\,\cos \,\,4x = \lambda$ has a solution, then $\lambda$ lies in the interval

The most general value of $\theta$ satisfying the equations $\sin \,\,\theta = \sin \,\,\alpha$ and $\cos \,\,\theta = \cos \,\,\alpha$ is

$\cos ^2 \frac{3 \pi}{5}+\cos ^2 \frac{4 \pi}{5}$ is equal to

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

The value of the expression $\left(\sqrt{3} \sin 75^{\circ}-\cos 75^{\circ}\right)$ is

Maximum value of the expression $2 \sin x+4 \cos x+3$ is

Sum of the root of the equation $2 \sin ^2 \theta+\sin ^2 2 \theta=2 ; 0 \leq \theta \leq \pi / 2$ is