General Solution of sin theta = sin alpha

Q.

The positive integer value of $n>3$ satisfying the equation $\frac{1}{\sin \left({\displaystyle \frac{\mathrm{\pi }}{n}}\right)}=\frac{1}{\sin \left({\displaystyle \frac{2\mathrm{\pi }}{n}}\right)}+\frac{1}{\sin \left({\displaystyle \frac{3\mathrm{\pi }}{n}}\right)}$ is

1. $8$

2. $6$

3. $5$

4. $7$

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

1. \N
2. 1
3. 2
4. infinite

Q. The general solution of the trigonometric equation sin x+ cos x =1 is given by

1. $x=2n\pi ;n=0,\pm 1,\pm \mathrm{2....}$
   
2. $x=2n\pi +\frac{\pi }{2};n=0,\pm 1,\pm 2,.....$
   
3. $x=n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{4}n=0,\pm 1,\pi 2,....$
   
4. $Noneofthese$

Q. If $\sin 2\theta \cos 2\theta =-\frac{1}{4},$ then the values of $\theta$ which satisfy the equation is

1. $\theta =\frac{n\pi }{8}+(-1{)}^n(-\frac{\pi }{48}),n\in Z$
2. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
3. $\theta =\frac{n\pi }{2}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
4. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{12}),n\in Z$

Q. How may values of $\theta ϵ[0,2\pi ]$ satisfies the equation $2cos\theta +sec\theta =5tan\theta$.
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Q. $Thegeneralsolutionoftheequation\sin \theta =\sin \alpha isgivenby\theta =n\pi +{(-1)}^n\alpha ,n\in Z.$

1. True
2. False

Q. $\mathrm{The}\mathrm{general}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{equation}2{\cos }^2\theta +3\sin \theta =0\mathrm{is}$
.

1. $n\pi +{(-1)}^n\frac{\pi }{6},n\in Z$
2. $n\pi +{(-1)}^{n+1}\frac{\pi }{6},n\in Z$
3. $n\pi +{(-1)}^n\frac{5\pi }{6},n\in Z$

Q. $\mathrm{The}\mathrm{general}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{equation}2{\cos }^2\theta +3\sin \theta =0\mathrm{is}$
.

1. $n\pi +{(-1)}^n\frac{\pi }{6},n\in Z$
2. $n\pi +{(-1)}^{n+1}\frac{\pi }{6},n\in Z$
3. $n\pi +{(-1)}^n\frac{5\pi }{6},n\in Z$

Q. Complete the set of values of $x$ in $(0,\pi )$ satisfying the inequation $1+{\log }_2\sin x+{\log }_2\sin 3x\ge 0$ is

1. $[0,\frac{\pi }{6}]$
2. $[\frac{\pi }{2},\frac{3\pi }{4}]$
3. $[\frac{\pi }{6},\frac{\pi }{4}]\cup [\frac{3\pi }{4},\frac{5\pi }{6}]$
4. $[\frac{5\pi }{6},\pi ]$

Q.

Set of values of $x$ in $(0,\pi )$ satisfying $1+{\log }_2\sin x+{\log }_2\sin 3x\ge 0$ is

1. $x\in (\frac{\pi }{2},\frac{2\pi }{3})$

2. $x\in [0,\frac{\pi }{3}]\cup [\frac{2\pi }{3},\pi ]$

3. $x\in (\frac{\pi }{3},\frac{2\pi }{3})$

4. $x\in [\frac{\pi }{6},\frac{\pi }{4}]\cup [\frac{3\pi }{4},\frac{5\pi }{6}]$

Q. The general values of $\theta$ satisfying the equation $2si{n}^2\theta -3sin\theta -2=0$ is $(n\in Z)$

1. $n\pi +(-1{)}^n\frac{\pi }{6}$
   
2. $n\pi +(-1{)}^n\frac{\pi }{2}$
   
3. $n\pi +(-1{)}^n\frac{5\pi }{6}$
   
4. $n\pi +(-1{)}^n\frac{7\pi }{6}$

Q.

The number of solutions of the pair of equations 2 $si{n}^2$θ - $co{s}^2$θ = 0 and 2 $si{n}^2$θ - 3 sin θ = 0, in the interval [0, 2π] is

1. Zero

2. One

3. Two

4. Four

Q. The general values of $\theta$ satisfying the equation $2si{n}^2\theta -3sin\theta -2=0$ is $(n\in Z)$

1. $n\pi +(-1{)}^n\frac{\pi }{6}$
   
2. $n\pi +(-1{)}^n\frac{\pi }{2}$
   
3. $n\pi +(-1{)}^n\frac{5\pi }{6}$
   
4. $n\pi +(-1{)}^n\frac{7\pi }{6}$

Q.

The number of solutions of the pair of equations 2 $si{n}^2$θ - $co{s}^2$θ = 0 and 2 $si{n}^2$θ - 3 sin θ = 0, in the interval [0, 2π] is

1. Zero

2. One

3. Two

4. Four

Q.

The positive integer value of n>3 satisfying the equation

$\frac{1}{sin\left(\frac{\pi }{n}\right)}=\frac{1}{sin\left(\frac{2\pi }{n}\right)}+\frac{1}{sin\left(\frac{3\pi }{n}\right)}$ is ___