iCON Education - Trigonometric Equations AP-EAPCET Test

If $\tan \left(\frac{\pi}{4}+\alpha\right)=\tan ^3\left(\frac{\pi}{4}+\beta\right)$, then $\tan (\alpha+\beta) \cot (\alpha-\beta)=$

$\sec ^2 2 \beta+\tan ^2 2 \beta$

$\operatorname{cosec}^2 2 \beta+\cot ^2 2 \beta$

$2\left(\sec ^2 2 \beta+\tan ^2 2 \beta\right)$

$4\left(\sec ^2 2 \beta+\tan ^2 2 \beta\right)$

If $0 \leq x \leq 3$ and $0 \leq y \leq 3$, then the number of solutions $(x, y)$ of the equation $\left(\sqrt{\sin ^2 x-\sin x+\frac{1}{2}}\right) 2^{\sec ^2 y}=1$ is

Statement I In the interval $[0,2 \pi]$, the number of common solutions of the equations $2 \sin ^2 \theta-\cos 2 \theta=0$ and $2 \cos ^2 \theta-3 \sin \theta=0$ is two.

Statement II The number of solutions of $2 \cos ^2 \theta-3 \sin \theta=0$ in $[0, \pi]$ is two.

Statement I and Statement II are both true

Statement I is true, Statement II is false

Statement I is false, Statement II is true

Statement I and Statement II are both false

The number of solutions of the equation $4 \cos 2 \theta \cos 3 \theta=\sec \theta$ in the interval $[0,2 \pi]$ is

$ \tan \frac{2 \pi}{7} \cdot \tan \frac{4 \pi}{7}+\tan \frac{4 \pi}{7} \cdot \tan \frac{\pi}{7}+\tan \frac{\pi}{7} \cdot \tan \frac{2 \pi}{7}= $

The sum of the solutions of $\cos x \sqrt{16 \sin ^2 x}=1$ in $(0,2 \pi)$ is

If $\sqrt{3} \cos \theta+\sin \theta>0$, then

$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$

$-\frac{\pi}{3}<\theta<\frac{2 \pi}{3}$

$-\frac{2 \pi}{3}<\theta<\frac{\pi}{3}$

$-\frac{\pi}{6}<\theta<\frac{5 \pi}{6}$

The general solution satisfying both the equations $\sin x=-\frac{3}{5}$ and $\cos x=-\frac{4}{5}$ is

$x=(2 n+1) \pi+\tan ^{-1}\left(\frac{3}{4}\right), n \in Z$

$x=2 n \pi+\tan ^{-1}\left(\frac{3}{4}\right), n \in Z$

$x=n \pi+\tan ^{-1}\left(\frac{3}{4}\right), n \in Z$

$x=n \pi \pm \tan ^{-1}\left(\frac{3}{4}\right), n \in Z$

The number of solutions of the equation $\sec x \cdot \cos 5 x+1=0$ in the interval $[0,2 \pi]$ is

If $2 \sin x-\cos 2 x=1$, then $\left(3-2 \sin ^2 x\right)=$

If $x \neq(2 n+1) \frac{\pi}{4}$, then the general solutions of $\cos x+\cos 3 x=\sin x+\sin 3 x$ is

$n \pi+\frac{\pi}{8}$

$n \pi \pm \frac{\pi}{8}$

$\frac{n \pi}{2} \pm \frac{\pi}{8}$

$\frac{n \pi}{2}+\frac{\pi}{8}$

The number of solutions of $\sin 2 x+\cos 4 x=2$ in the interval $[-\pi, \pi]$ is

Number of solutions of the equation $\cos \theta+\cos 2 \theta-\sqrt{3}(\sin \theta+\sin 2 \theta)+1=0$ lying in the interval $(0,2 \pi)$ is

The number of solutions of the equation $2 \sin ^2 \theta-3 \cos ^2 \theta=\sin \theta \cos \theta$ lying in the intervals $(-\pi, \pi)$ is

The general solution of $\cot \frac{x}{2}-\cot x=\operatorname{cosec} \frac{x}{2}$ is

$\left\{\left.2 n \pi+\frac{\pi}{3} \right\rvert\, n \in z\right\}$

$\left\{\left.4 n \pi+\frac{\pi}{3} \right\rvert\, n \in z\right\}$

$\left\{\left.2 n \pi+\frac{2 \pi}{3} \right\rvert\, n \in z\right\}$

$\left\{\left.4 n \pi \pm \frac{2 \pi}{3} \right\rvert\, n \in z\right\}$

The values of $x$ in $(-\pi, \pi)$, which satisfy the equation $8^{1+\cos ^2 x+\cos ^4 x+\ldots \ldots}=4^3$ are

$\pm \frac{\pi}{4}, \pm \frac{3 \pi}{4}$

$\pm \frac{\pi}{6}, \frac{\pi}{3}$

The general solution of the equation $\tan x+\tan 2 x-\tan 3 x=0$ is

$\left\{x \left\lvert\, x=n \pi \pm \frac{\pi}{3}\right.\right.$ or $\left.\frac{n \pi}{2}, n \in Z\right\}$

$\left\{x \left\lvert\, x=n \pi \pm \frac{\pi}{3}\right.\right.$ or $\left.n \pi, n \in Z\right\}$

$\left\{x \left\lvert\, x=n \pi \pm \frac{\pi}{3}\right.\right.$ or $\frac{n \pi}{2}$ or $\left.n \pi, n \in Z\right\}$

$\left\{x \left\lvert\, x=n \pi \pm \frac{\pi}{6}\right.\right.$ or $\left.\frac{n \pi}{2}, n \in Z\right\}$

Number of solutions of the trigonometric equation $2 \tan 2 \theta-\cot 2 \theta+1=0$ lying in the interval $[0, \pi]$ is

For $a \in R-\{0\}$, if $a \cos x+a \sin x+a=2 k+1$ has a solution, then $k$ lies in the interval

$\left[\frac{a-1-\sqrt{2} a}{2}, \frac{a-1+\sqrt{2} a}{2}\right]$

$\left[\frac{a+1-\sqrt{2}}{2}, \frac{a+1+\sqrt{2}}{2}\right]$

$\left[\frac{a-1-\sqrt{2}}{2}, \frac{a-1+\sqrt{2}}{2}\right]$

$\left[-\frac{\left(\sqrt{2 a^2+2 a+1}+1\right)}{2}, \frac{\left(\sqrt{2 a^2+2 a+1}-1\right)}{2}\right]$

If the general solution .set of $\sin x+3 \sin 3 x+\sin 5 x=1$ is $S$, then $\{\sin \alpha / \alpha \in S\}=$

$\left\{\frac{1}{2}, \frac{-1}{2}, 0,1,-1\right\}$

$\left\{\frac{\sqrt{3}}{2}, 0, \frac{-\sqrt{3}}{2}\right\}$

$\left\{1,-1, \frac{\sqrt{3}}{2}, 0, \frac{-\sqrt{3}}{2}\right\}$

The general solution of the equation $\sin ^2 \theta+3 \cos ^2 \theta=$ $5 \sin \theta$ is

$n \pi \pm \frac{\pi}{3}, n \in Z$

$n \pi+(-1)^n \frac{\pi}{6}, n \in Z$

$n \pi \pm \frac{\pi}{6}, n \in Z$

$n \pi+(-1)^n \frac{\pi}{3}, n \in Z$

The number of ordered pairs $(x, y)$ satisfying the equations $\sin x+\sin y=\sin (x+y)$ and $|x|+|y|=1$ is

If $5 \sin h x-\cos h x=5$, then one of the values of $\tan h x$ is

The smallest positive value (in degrees) of $\theta$ for which $\tan \left(\theta+100^{\circ}\right)=\tan \left(\theta+50^{\circ}\right) \tan (\theta) \tan \left(\theta-50^{\circ}\right)$ is valid, is

The general solution of

$ \begin{aligned} & 4 \cos 2 x-4 \sqrt{3} \sin 2 x+\cos 3 x-\sqrt{3} \sin 3 x \\ & \qquad+\cos x-\sqrt{3} \sin x=0 \end{aligned} $

$\frac{n \pi}{2}-\frac{\pi}{3}$

$\frac{n \pi}{2}+\frac{\pi}{6}$

$\frac{n \pi}{2}+\frac{\pi}{12}$

$\frac{n \pi}{2}-\frac{\pi}{12}$

The general solution of $2 \cos ^2 x-2 \tan x+1=0$ is

$n \pi+\frac{\pi}{4}, n \in Z$

$2 n \pi \pm \frac{\pi}{4}, n \in Z$

$2 n \pi \pm \frac{\pi}{3}, n \in Z$

$n \pi \pm \frac{\pi}{3}, n \in Z$

$1+\sin x+\sin ^2 x+\sin ^3 x+\ldots \ldots+\infty=4+2 \sqrt{3}$ and $0

$\frac{\pi}{6}$

$\frac{5\pi}{6}$

$\frac{2\pi}{3}$

$\frac{7\pi}{3}$

$\text { If } \sin \theta+\operatorname{cosec} \theta=4, \text { then } \sin ^2 \theta+\operatorname{cosec}^2 \theta=$

If $2 \cosh 2 x+10 \sinh 2 x=5$, then $x=$

$\frac{1}{2} \log \frac{4}{3}$

$\frac{1}{2} \log \frac{2}{3}$

$\frac{1}{2} \log \frac{3}{2}$

$\frac{1}{2} \log \frac{3}{4}$

If $\sin \left(\frac{\pi}{4} \cos \theta\right)=\cos \left(\frac{\pi}{4} \tan \theta\right)$, then $\theta$ is equal to

$2 n \pi+\frac{\pi}{4}$

$2 n \pi \pm \frac{\pi}{4}$

$2 n \pi-\frac{\pi}{4}$

$n \pi+\frac{\pi}{4}$

If $\theta \in[0,2 \pi]$ and $\cos 2 \theta=\cos \theta+\sin \theta$, then the sum of all values of $\theta$ satisfying the equation is

The value of $x$ satisfying the equation $3 \operatorname{cosec} x=4 \sin x$ are

$\frac{\pi}{6}, \frac{\pi}{3}$

$\frac{\pi}{3}, \frac{\pi}{4}$