Principal Solution of Trigonometric Equation

Q. The value of $\sin 1^{\circ }+\sin 2^{\circ }+\cdots +\sin {359}^{\circ }$ is equal to

1. $0$
2. $1$
3. $180$
4. $-1$

Q. Find the principal value of ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right).$

Q.

The domain of ${\sin }^{-1}x$ is

1. $\left(-\mathrm{\pi },\mathrm{\pi }\right)$

2. $\left[-1,1\right]$

3. $\left(0,2\mathrm{\pi }\right)$

4. $\left(-\infty ,\infty \right)$

Q.

What is $\sec 180°$ degrees?

Q. If $f\left(x\right)=x^3+3x^2+4x+asinx+bcosx\forall xϵR$ is one - one function then the maximum value of $a^2+b^2$ is

1. $\sqrt{3}$
2. 4
3. 1
4. 2

Q.

The function $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$ increases, if

1. $0<x<\frac{\mathrm{\pi }}{8}$

2. $\frac{\mathrm{\pi }}{4}<x<\frac{3\mathrm{\pi }}{8}$

3. $\frac{3\mathrm{\pi }}{8}<x<\frac{5\mathrm{\pi }}{8}$

4. $\frac{5\mathrm{\pi }}{8}<x<\frac{3\mathrm{\pi }}{4}$

Q. Prove that: $\tan 4x=\frac{4\tan x(1-{\tan }^{2}x)}{1-6{\tan }^{2}x+{\tan }^{4}x}$

Q. Let the solution curve $y=y\left(x\right)$ of the differential equation $(1+e^{2x})(\frac{dy}{dx}+y)=1$ pass through the point $(0,\frac{\pi }{2})$, Then $\underset{x\to \infty }{lim}{e}^{x}y\left(x\right)$ is equal to

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

Q. The number of solutions of the equation $\cos \theta +\cos 3\theta +\cos 5\theta +\cos 7\theta =0,$ where $\theta \in [0,2\pi ]$ is

Q. If $\alpha$ is one of the principal solutions which satisfies the equation $1+{\sin }^2\theta =3\sin \theta \cos \theta ,$ then which of the following is not possible?

1. $\sin 2\alpha =\frac{4}{5}$
2. $\cos 2\alpha =0$
3. $\sin 3\alpha =\frac{1}{\sqrt{2}}$
4. $\cos 3\alpha =\frac{2}{\sqrt{5}}$

Q.

Solve:

$\sin 3x+\cos 2x=0$

Q. The number of distinct solutions of $\sec x+\tan x=\sqrt{3}$, where $0\le x\le 3\pi$ is

Q. $\mathrm{The}\mathrm{principle}\mathrm{solutions}\mathrm{of}\mathrm{the}\mathrm{trigonometric}\mathrm{equation}\sec \theta =\frac{2}{\sqrt{3}}\mathrm{are}$

1. 
   $\frac{\pi }{6}$
2. 
   $-\frac{\pi }{6}$
3. 
   $\frac{11\pi }{6}$
4. 
   $\frac{13\pi }{6}$

Q. Number of solutions of the equation $2\sin x-2\sqrt{3}\cos x-\sqrt{3}\tan x+3=0$ where $x\in [0,2\pi )$ is

Q. The number of solution(s) of the equation ${\tan }^{2}x-{\sec }^{10}x+1=0$ in $x\in [0,60]$ is

Q. If $\alpha ,\beta$ are the solutions of $\sin x=-\frac{1}{2}$ in $[0,2\pi ]$ and $\alpha ,\gamma$ are the solutions of $\cos x=-\frac{\sqrt{3}}{2}$ in $[0,2\pi ],$ then

1. $\alpha -\beta =\pi$
2. $\beta -\gamma =\pi$
3. $\alpha -\gamma =\pi$
4. $\alpha +\beta =3\pi$

Q. If $0\le x\le 3\pi ,0\le y\le 3\pi$ and $\cos x\sin y=1$, then the possible number of values of the ordered pair $(x,y)$ is

1. $4$
2. $5$
3. $6$
4. $7$

Q. If principal argument of $z_0$ satisfying $|z-3|\le \sqrt{2}$ and $\arg (z-5i)=-\frac{\pi }{4}$ simultaneously is $\theta ,$ then the CORRECT statement(s) is/are

1. $|z_0|=17$
2. $\tan 2\theta =\frac{8}{15}$
3. $\tan \theta =-\frac{1}{4}$
4. $|z_0-5i|=4\sqrt{2}$

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

1. $n\pi +\frac{\pi }{6},n\in Z$
2. $2n\pi +\frac{\pi }{3},n\in Z$
3. $n\pi +\frac{\pi }{3},n\in Z$
4. no solution

Q. The value's of $x\in (0,\frac{\pi }{2})$ satisfying $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$ are

1. $\frac{\pi }{12}$
2. $\frac{5\pi }{12}$
3. $\frac{7\pi }{24}$
4. $\frac{11\pi }{36}$

Q.

$\sqrt{\frac{sec\theta -1}{sec\theta +1}}+\sqrt{\frac{sec\theta +1}{sec\theta -1}}=2\cos ec\theta$

Q. The number of solution(s) of the equation $\tan x\tan 4x=1$ for $0<x<\pi$ is

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

Q. Prove the following :
$2{\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)={\tan }^{-1}\left(\frac{31}{17}\right)$

Q. If $\sin \theta +\sqrt{3}\cos \theta =6x-x^2-11$, where $0\le \theta \le 4\pi ,x\in R$, then which of the following options is/are correct ?

1. ${\theta }_{min}=\frac{7\pi }{6}$
2. ${\theta }_{max}=\frac{19\pi }{6}$
3. There are two pairs of values for $(x,\theta )$
4. There are four pairs of values for $(x,\theta )$

Q. If $x+y=\frac{2\pi }{3}$ and $\frac{\sin x}{\sin y}=2$, then which of the following option(s) is/are correct?

1. there are $4$ values of $x$ in $[0,4\pi ]$ for $y\in R-\left\{n\pi \right\}$
2. there are $6$ values of $x$ in $[0,4\pi ]$ for $y\in R-\left\{n\pi \right\}$
3. there are $6$ values of $y$ in $[0,4\pi ]$ for $x\in R$
4. there are $4$ values of $y$ in $[0,4\pi ]$ for $x\in R$

Q.

The number of real solutions of $\sin \left(e^x\right)·\cos \left(e^x\right)=2^{x-2}+2^{-x-2}$ is

1. $0$

2. $1$

3. $2$

4. $\infty$

Q. If $x\in [0,100\pi ]$, then the number of solutions of the equation ${\sin }^{4}x-{\cos }^{2}x\sin x+2{\sin }^{2}x+\sin x=0$ is

Q. Number of solutions of the equation $|\sin 4\theta |=1$ in $\theta \in [0,2\pi ]$ is

Q. The number of values of $x$ in $[0,2\pi ]$ which satisfy $\tan x+\tan 4x+\tan 7x=\tan x\tan 4x\tan 7x$ is

Q. The number of solutions of ${16}^{{\sin }^{2}x}+{16}^{{\cos }^{2}x}=10$ in the interval $x\in [0,\pi ]$ is

1. $4$
2. $0$
3. $2$
4. $8$