Trigonometric Equations

Q.

$1+\cos 2x+\cos 4x+\cos 6x=$

1. $2\cos x\cos 2x\cos 3x$

2. $4\sin x\cos 2x\cos 3x$

3. $4\cos x\cos 2x\cos 3x$

4. $Noneofthese$

Q. An equation involving one or more trigonometric ratios of unknown angle is called a trigonometric equation.

1. True
2. False

Q.

The value of the $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}+2\mathrm{sinx}}{\sqrt{{\mathrm{x}}^2+2\mathrm{sinx}+1}-\sqrt{\mathrm{x}-{\sin }^2\mathrm{x}+1}}$is

Q. The number of solutions of the equation $1+\sin x{\sin }^2\frac{x}{2}=0,$ in $[-\pi ,\pi ]$

1. zero
2. one
3. two
4. three

Q. Let $\alpha$ and $\beta$ be non zero real numbers such that $2(cos\beta -cos\alpha )+cos\alpha cos\beta =1$. Then which of the following is/are true?

1. $\sqrt{3}tan\left(\frac{\alpha }{2}\right)-tan\left(\frac{\beta }{2}\right)=0$
2. $tan\left(\frac{\alpha }{2}\right)-\sqrt{3}tan\left(\frac{\beta }{2}\right)=0$
3. $tan\left(\frac{\alpha }{2}\right)+\sqrt{3}tan\left(\frac{\beta }{2}\right)=0$
4. $\sqrt{3}tan\left(\frac{\alpha }{2}\right)+tan\left(\frac{\beta }{2}\right)=0$

Q. The total number of solutions of tanx+cotx=2cosecx in $[-2\pi ,2\pi ]$ is equal to.

1. 7
2. 6
3. 4
4. 2

Q.

Least positive integral value of x satisfying
$(e^x-2)(sinx-cosx)(x-lo{g}_{e}2)(cosx-\frac{1}{\sqrt{2}})<0$ is

1. 5

2. 2

3. 4

4. 3

Q. If $cosx-sin\alpha cot\beta sinx=cos\alpha$ ,
then the value of $tan\left(\frac{x}{2}\right)$ is

1. $-tan\left(\frac{\alpha }{2}\right)cot\left(\frac{\beta }{2}\right)$
2. $tan\left(\frac{\alpha }{2}\right)tan\left(\frac{\beta }{2}\right)$
3. $-cot\left(\frac{\alpha }{2}\right)tan\left(\frac{\beta }{2}\right)$
4. $cot\left(\frac{\alpha }{2}\right)tan\left(\frac{\beta }{2}\right)$

Q. If $\alpha$ is a root of equation $(2\sin x-\cos x)(1+\cos x)={\sin }^{2}x$, $\beta$ is a root of equation $3{\cos }^{2}x-10\cos x+3=0$ and $\gamma$ is a root of equation $1-\sin 2x=\cos x-\sin x;0\le \alpha ,\beta ,\gamma \le \frac{\pi }{2},$ then $\cos \alpha +\cos \beta +\cos \gamma$ can be equal to

1. $\frac{3\sqrt{6}+2\sqrt{2}+6}{6\sqrt{2}}$
2. $\frac{3\sqrt{3}-8}{6}$
3. $\frac{3\sqrt{3}+2}{6}$
4. None of these

Q. $co{t}^{-1}\left[\right(cos\alpha {)}^{\frac{1}{2}}]-ta{n}^{-1}[\left(cos\alpha {)}^{\frac{1}{2}}\right]=x$, then $sinx=$

1. $tan\alpha$
2. $ta{n}^2\left(\frac{\alpha }{2}\right)$
3. $co{t}^2\left(\frac{\alpha }{2}\right)$
4. $cot\left(\frac{\alpha }{2}\right)$

Q.

How do you create a cosine equation?

Q. The inequality $2\cos x\le |\sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}|\le \sqrt{2}$ where $x\in [0,2\pi ]$ holds true in the interval $[\frac{a\pi }{2},\frac{b\pi }{2}]$. Then the value of $a+b$ is

Q. Let $\cos A+\cos B=x$; $\cos 2A+\cos 2B=y$; $\cos 3A+\cos 3B=z$, then which of the following is true?

1. ${\cos }^{2}A+{\cos }^{2}B=1+\frac{y}{2}$
2. $\cos A.\cos B=\frac{1}{4}(2x^2-y-2)$
3. $2x^3+z=3x(1+y)$
4. $xyz=0,\forall A,B\in R$

Q.

$e^{|sinx|}+e^{-|sinx|}+4a=0$ will have exactly four different solutions in $[0,2\pi ]$ if

1. $a\in R$
   

2. $a\in [\frac{-e}{4},\frac{-1}{4}]$
   

3. $a\in \left[\frac{-1-e^2}{4e}\right],\infty$
   

4. None

Q. If $x+\sin y=2020$ and $x+2020\cos y=2019$, where $0\le y\le \frac{\pi }{2},$ then the value of $[x+y]$ is
( $[.]$ denotes the greatest integer function )

Q. Number of real solutions of the equation $\sin \left(e^x\right)=5^x+5^{-x}$ is

1. $0$
2. $1$
3. $2$
4. infinitely many

Q.

The number of distinct solutions of the equation
$\frac{5}{4}co{s}^{2}2x+co{s}^{4}x+si{n}^{4}x+co{s}^{6}x+si{n}^{6}x=2$
In the interval $[0,2\pi ]$ is ___

Q. If $\tan 2\alpha \cdot \tan \beta =1$ and $\tan \alpha \cdot \tan \gamma =1$ then

1. $(1+\tan \alpha )(1+\tan \left(\frac{\beta }{2}\right))=2\sin (\alpha +\gamma )$
2. $\tan \alpha +\tan \beta =\tan \gamma$
3. $\tan (\alpha +\beta )=\tan \gamma$
4. $\tan 2\gamma \tan \beta =-1$

Q. If $x+\sin y=2020$ and $x+2020\cos y=2019$, where $0\le y\le \frac{\pi }{2},$ then the value of $[x+y]$ is
([.] denotes greatest integer function )

Q.

Total number of values in $(-2\pi ,2\pi )$ and satisfying
$lo{g}_{|cosx|}|sinx|+lo{g}_{|sinx|}|cosx|=2$ is

1. 2

2. 4

3. 6

4. 8

Q. The value of an unknown angle which satisfies the given trigonometric equation is called a solution or a root of the equation.

1. False
2. True

Q. If $\alpha$ is a root of equation $(2\sin x-\cos x)(1+\cos x)={\sin }^{2}x$, $\beta$ is a root of equation $3{\cos }^{2}x-10\cos x+3=0$ and $\gamma$ is a root of equation $1-\sin 2x=\cos x-\sin x;0\le \alpha ,\beta ,\gamma \le \frac{\pi }{2},$ then $\sin (\alpha -\beta )$ is equal to

1. $\frac{1-\sqrt{6}}{6}$
2. $\frac{1-2\sqrt{3}}{6}$
3. $\frac{1-2\sqrt{6}}{6}$
4. $\frac{\sqrt{3}-2\sqrt{2}}{6}$

Q. The most general solutions of the equation $\sec x-1=(\sqrt{2}-1)\tan x$ are given by

1. $n\pi +\frac{\pi }{8}$
2. $2n\pi ,2n\pi +\frac{\pi }{4}$
3. $2n\pi$
4. None of these

Q.

Complete set of values of x satisfying cos2x > |sinx|, $x\in (\frac{-\pi }{2},\pi )$ is

1. $(\frac{-\pi }{6},\pi )$

2. $(\frac{-\pi }{6},\frac{\pi }{6})\cup (\frac{5\pi }{6},\pi )$

3. $(\frac{\pi }{6},\frac{\pi }{2})\cup (\frac{5\pi }{6},\pi )$

4. $(\frac{-\pi }{6},\frac{5\pi }{6})$

Q.

$e^{|sinx|}+e^{-|sinx|}+4a=0$ will have exactly four different solutions in $[0,2\pi ]$ if

1. $a\in R$
   

2. $a\in [\frac{-e}{4},\frac{-1}{4}]$
   

3. $a\in \left[\frac{-1-e^2}{4e}\right],\infty$
   

4. None