Trigonometric Equations

Q.

Find tan ${15}^{\circ }$ and show that tan ${15}^{\circ }$ + cot ${15}^{\circ }$ = 4

Q.

Find the number of solutions of $sinx=\left(\frac{x}{10}\right)$.

___

Q.

Evaluate tan $\frac{13\pi }{12}$

Q. $\frac{\tan \left(\frac{35\pi }{6}\right).\sin \left(\frac{11\pi }{3}\right).\sec \left(\frac{7\pi }{3}\right)}{\cot \left(\frac{5\pi }{4}\right).cosec\left(\frac{7\pi }{4}\right).\cos \left(\frac{17\pi }{6}\right)}=$

1. 
2. 
3. 
4. 

Q. $2si{n}^2\left(\left(\frac{\pi }{2}\right)co{s}^{2}x\right)=1-cos\left(\pi sin2x\right),x\ne (2n+1)\frac{\pi }{2},nϵI$, then $cos2x$ is equal to

1. $\frac{1}{5}$
2. $\frac{3}{5}$
3. $\frac{4}{5}$
4. $1$

Q.

The set of all x in $(-\pi ,\pi )$ satisfying $|4sinx-1|<\sqrt{5}$ is given by

1. 

2. 

3. 

4. 

Q.

The number of solutions of equation [sin x ] = [1 + sin x] + [ 1 - cos x] where [.] denotes the greatest integer function and is

1. 2 solutions

2. 3 solutions

3. No solution

4. 1 solution

Q.

Find the value of the expression $co{s}^4\frac{\pi }{8}+co{s}^4\frac{3\pi }{8}+co{s}^4\frac{5\pi }{8}+co{s}^4\frac{7\pi }{8}$

Or

Find the value of $sin\frac{x}{2},cos\frac{x}{2}$ and $tan\frac{x}{2}$, if $sinx=\frac{1}{4}$, where x lies in ssecond quadrant.

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 sin x + cos x = t, then sin 3x - cos 3x is equal to

1. t (2 - 3)

2. 3t (2 + 1)

3. t (2 + 3)

4. -2t + 1)

Q. A trigonometric equation holds true for value(s) of the argument

1. all
2. some
3. exactly one

Q.

How do you solve an equation with 2 trigonometric functions?

Q.

If $2tan\frac{\alpha }{2}=tan\frac{\beta }{2},$ prove that $cos\alpha =\frac{3+5cos\beta }{5+3cos\beta }.$ A.so, determine the value of $cos\alpha$ when $\beta ={60}^{\circ }.$

Or

If a $sin\theta =bsin(\theta +\frac{2\pi }{3})=csin(\theta +\frac{4\pi }{3}),$ find the value of ab + bc + ca.

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. 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.

$Let\alpha ,\beta besuchthat\pi <\alpha -\beta <3\pi .Ifsin\alpha +sin\beta =\frac{-21}{65}andcos\alpha +cos\beta =\frac{-27}{65},thenthevalueofcos\frac{\alpha -\beta }{2}is$

1. 

2. 

3. 6/65

4. -6/65

Q.

Find the general solution of the equation 4 sin x cos x + 2 sin x + 2 cos x +1 = 0.

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. 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.

Trigometric identities are valid for all real values where as trigonometric equations are valid for only some values.

1. 1
2. 2

Q.

The value of ${\prod }_{k=1}^6(sin\frac{2\pi k}{7}-icos\frac{2\pi k}{7})$ is:

1. 0

2. -1

3. -i

4. i

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 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. 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. Find the derivatives of the following:
$5\sin x-6\cos x+7$.

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