General Solution of Sin Theta = Sin Alpha

Q.

The general solution of the trigonometric equation $\sin x+\cos x=1$ is given by:

1. $x=2n\pi ;n=0,\pm 1,\pm 2...$

2. $x=2n\pi +\pi ;n=0,\pm 1,\pm 2...$

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

4. none of these

Q.

Find the modulus and the arguments of each of the complex numbers

$z=-1-i\sqrt{3}$

Q. Which one of the following equations has no solution?

1. 
2. 
3. 
4. 

Q.

If x = h + a sec $\theta$ and y = k + b cos ec $\theta$, Then

1. 

2. 

3. 

4. 

5. x² + y² = a² + b²

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 set of values of $\theta$ satisfying the inequation $2si{n}^2\theta -5sin\theta +2>0$, where 0<$\theta$<2$\pi$, is

1. 

2. 

3. 

4. 

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

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

Q.

For any complex numbers $z,z_1$ and $z_2$ prove that:
(i) $arg\left(\overline{z}\right)=-arg\left(z\right)$
(ii) $arg\left(z_1{z}_2\right)=arg\left(z_1\right)+arg\left(z_2\right)$
(iii) $arg\left(z_1\overline{z_2}\right)=arg\left(z_1\right)-arg\left(z_2\right)$
(iv) $arg\left(\frac{z_1}{z_2}\right)=arg\left(z_1\right)-arg\left(z_2\right)$

Q. How may values of $\theta ϵ[0,2\pi ]$ satisfies the equation $2cos\theta +sec\theta =5tan\theta$.
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Q.

Find the principal solutions of each of the following equations.

(i)sin $x=\frac{1}{2}$

(ii)cos $x=\frac{1}{\sqrt{2}}$

(iii)tan $x=\frac{1}{\sqrt{3}}$

Q.

If r > 0, -$\pi$ ≤ $\theta$ ≤ $\pi$ and (r, $\theta$) satisfy r sin$\theta$ = 3 and r = 4(1 + sin$\theta$) then the number of possible solutions of the pair ( r, $\theta$) is

1. 2

2. 4

3. 0

4. Infinite

Q.

Find the values of $\theta$ and p, if the equation $xcos\theta +ysin\theta =p$ is the normal form of the line $\sqrt{3}x+y+2=0$

Q.

Find the value $si{n}^4\theta$ - $co{s}^4\theta$ + 2$co{s}^2\theta$, when $\theta =\frac{\Pi }{7}$

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Q.

for all values of $\theta$ , the values of $3-cos\theta +cos(\theta +\frac{\pi }{3})$ lie in the interval

1. [-2, 3]

2. [-2, 1]

3. [2, 4]

4. [1, 5]

Q.

Prove that $sinx+sin7x+sin3x+sin5x=4cosxcos2xsin4x$

Or

Solve $\sqrt{3}cosx-sinx=1$

Q. The general solution of the equation $\sin x=\frac{\sqrt{3}}{2}$ is x=

1. 2nπ±π3
2. nπ+(-1)nπ3
3. nπ ± π6

Q. Find the equation of the parabola that satisfies the following conditions: Vertex $(0,0)$, focus $(-2,0)$

Q.

1) $cos2x=co{s}^{2}x-si{n}^{2}x$

2) $1+sin2x=(cosx+sinx{)}^2$

3) $1-sin2x=(cosx-sinx{)}^2$

How many of the above statements are correct?

Q. If $\sin 3\theta =\sin \theta ,$ how many solutions exist such that $-2\pi <\theta <2\pi$ ?

1. 8
2. 9
3. 5
4. 7

Q.

Find the value of sin 13x + sin x if cos 12x = cos 14 x

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Q.

Find the value of sin 13x + sin x if cos 12x = cos 14 x

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Q. If $\alpha ,\beta$ are two values of $\theta$ satisfying the equarion $\frac{\cos \theta }{a}+\frac{\sin \theta }{a}=\frac{1}{c},$ then prove -
$\frac{\cos \left(\frac{\alpha +\beta }{2}\right)}{\cos \left(\frac{\alpha -\beta }{2}\right)}=\frac{c}{a}$