Solving Simultaneous Trigonometric Equations

Q.

How do I find the value of $\sin 150°$.

Q. If the angle of elevation of a cloud from a point $P$ which is $25\text{m}$ above a lake be ${30}^{\circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be ${60}^{\circ }$, then the height of the cloud (in meters) from the surface of the lake is :

1. $60$
2. $50$
3. $45$
4. $42$

Q. The value of $\tan (2{\tan }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right))$ is equal to:

1. $\frac{220}{21}$
2. $\frac{151}{63}$
3. $\frac{-181}{69}$
4. $\frac{-291}{76}$

Q. Let $a,b,c$ be three non zero real numbers such that the equation
$\sqrt{3}a\cos x+2b\sin x=c,x\in [-\frac{\pi }{2},\frac{\pi }{2}]$
has two distinct real roots $\alpha$ and $\beta$ with $\alpha +\beta =\frac{\pi }{3}$. Then, the value of $\frac{b}{a}$ is .

Q.

Show that ${\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(\frac{2}{11}\right)={\tan }^{-1}\left(\frac{3}{4}\right)$

Q.

The number of all possible values of $\theta ,where0<\theta <\pi$, for which the system of equations $(y+z)cos3\theta =\left(xyz\right)sin3\theta$
$xsin3\theta =\frac{2cos3\theta }{y}+\frac{2sin3\theta }{z}\left(xyz\right)sin3\theta =(y+2z)cos3\theta +ysin3\theta$
have a solution $(x_0,y_0,z_0)with{y}_0{z}_0\ne 0$ is___

Q. The sum of all values of $\theta \in (0,\frac{\pi }{2})$ satisfying ${\sin }^{2}2\theta +{\cos }^{4}2\theta =\frac{3}{4}$ is :

1. $\pi$
2. $\frac{\pi }{2}$
3. $\frac{3\pi }{8}$
4. $\frac{5\pi }{4}$

Q. The number of all the possible triplets $a_1,a_2,a_3$ such that $a_1+a_2\cos \left(2x\right)+a_3{\sin }^2\left(x\right)=0$ for all $x\in R$ is

1. $0$
2. $1$
3. $3$
4. infinite

Q. The number of integral solution(s) of ${\cos }^{-1}(4x^3-12x^2+11x-\frac{5}{2})=\frac{\pi }{3}$ is

1. $0$
2. $1$
3. $2$
4. $3$

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 integral values of ‘$k$’ for which the equation $3\sin x+4\cos x=k+1$ has a solution, $k\in R$ is

Q. If $sin\alpha +sin\beta =a$ and $cos\alpha +cos\beta =b,$ show that
(i)$sin(\alpha +\beta )=\frac{2ab}{a^2+b^2}$
(ii)$cos(\alpha +\beta )=\frac{b^2-a^2}{b^2+a^2}$

Q. The number of solutions of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$, where $x\in [-\pi ,\pi ]$ is

1. $7$
2. $8$
3. $6$
4. $4$

Q. General solution of the equation $2{\cot }^{2}x={\text{cosec}}^{2}x$ is :

1. $n\pi \pm \frac{\pi }{6},n\in Z$
2. $n\pi \pm \frac{\pi }{2},n\in Z$
3. $n\pi \pm \frac{3\pi }{4},n\in Z$
4. $n\pi \pm \frac{\pi }{4},n\in Z$

Q. The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be ${45}^{\circ }$ from a point $A$ on the plane. Let $B$ be the point $30$ m vertically above the point $A$. If the angle of elevation of the top of the tower from $B$ be ${30}^{\circ }$, then the distance (in m) of the foot of the tower from the point $A$ is :

1. $15(3-\sqrt{3})$
2. $15(3+\sqrt{3})$
3. $15(1+\sqrt{3})$
4. $15(5-\sqrt{3})$

Q. If ${\sin }^{2}x-2\sin x-1=0$ has exactly four different solutions in the interval $[0,n\pi ]$ , then possible value(s) of $n$ is/are :

1. $5$
2. $3$
3. $4$
4. $6$

Q. What is the maximum distance of the normal drawn from a variable point of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ from the centre of the ellipse.

1. $2$ if $a=5,b=3$
2. $10$ if $a=5,b=3$
3. $2$ if $a=6,b=4$
4. $10$ if $a=6,b=4$

Q. The general solution of $\sqrt{3}\tan \theta -1=0$ is :

1. $\theta =n\pi +\frac{\pi }{3};n\in Z$
2. $\theta =n\pi +\frac{\pi }{6};n\in Z$
3. $\theta =2n\pi +\frac{\pi }{6};n\in Z$
4. $\theta =2n\pi +\frac{\pi }{3};n\in Z$

Q. If $\left[\sin x\right]+\left[\sqrt{2}\cos x\right]=-3,x\in [0,2\pi ]$, (where $[.]$ represents the greatest integer function), then the range of $x$ is

1. $(\frac{5\pi }{4},\frac{7\pi }{4})$
2. $(\frac{3\pi }{4},\frac{5\pi }{4})$
3. $(\pi ,\frac{5\pi }{4})$
4. $(\frac{\pi }{4},\frac{7\pi }{4})$

Q. The number of solutions of $\sin 3x=\cos 2x,$ in the interval $(\frac{\pi }{2},\pi )$ is :

1. $4$
2. $1$
3. $2$
4. $3$

Q. Total number of solutions of $\tan x+\cot x=2\text{cosec}x$ , where $x\in [-2\pi ,2\pi ]$ is :

1. $5$
2. $4$
3. $6$
4. $3$

Q.

Find the value of $\theta$ satisfying $\left|\begin{array}{ccc}1& 1& sin3\theta \\ -4& 3& cos2\theta \\ 7& -7& -2\end{array}\right|$ = 0

Q. Total number of solution of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$ in $x\in [-\pi ,\pi ]$ is :

1. $6$
2. $4$
3. $8$
4. $7$

Q. If ${\tan }^{2}x=1$ , then $x$ is equal to

1. $2n\pi +(-1{)}^n\frac{\pi }{4},n\in Z$
2. $n\pi \pm \frac{\pi }{4},n\in Z$
3. $2n\pi +\frac{\pi }{4},n\in Z$
4. $n\pi +\frac{\pi }{4},n\in Z$

Q. The number of solutions of $\sin 3x=\cos 2x,$ in the interval $(\frac{\pi }{2},\pi )$ is :

1. $4$
2. $1$
3. $3$
4. $2$

Q. prove that $\frac{\sin (x+y)}{\sin (x-y)}=\frac{\tan x+\tan y}{\tan x-\tan y}$

Q. Prove that: $\frac{\sin x-\sin y}{\cos x+\cos y}=\tan \frac{x-y}{2}$

Q. Let $A,B,C$ are three angles such that
$\sin A+\sin B+\sin C=0,$ then the value of $\frac{\sin A.\sin B.\sin C}{\sin 3A+\sin 3B+\sin 3C}$ (wherever defined) is

1. $-12$
2. $12$
3. $-\frac{1}{12}$
4. $\frac{1}{12}$

Q.

The number of solution of cos$\theta$ + $\sqrt{3}$sin$\theta$ = 5, 0 ≤ $\theta$ ≤ 5$\pi$ is

1. 5

2. 1

3. 4

4. 0

Q. Prove that
${\tan }^{-1}\sqrt{x}=\frac{1}{2}{\cos }^{-1}\left(\frac{1-x}{1+x}\right),x\in [0,1]$