Principal Solution of Trigonometric Equation

Q. The principal solution of the equation $\sin x=\frac{1}{2}$ that is less than $\frac{\pi }{2}$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{6}$

Q. Let $f:R\to R$ be defined as $f(x+y)+f(x-y)=2f\left(x\right)f\left(y\right)$, $f\left(\frac{1}{2}\right)=-1$. Then the value of $\sum _{k=1}^{20}\frac{1}{\sin \left(k\right)\sin (k+f(k\left)\right)}$ is equal to

1. ${\text{cosec}}^2\left(21\right)\cos \left(20\right)\cos \left(2\right)$
2. ${\sec }^2\left(21\right)\sin \left(20\right)\sin \left(2\right)$
3. ${\sec }^2\left(1\right)\sec \left(21\right)\cos \left(20\right)$
4. ${\text{cosec}}^2\left(1\right)\text{cosec}\left(21\right)\sin \left(20\right)$

Q.

A conic section is defined by the equations x = - 1 + sect y = 2 + z tant. The coordinates of the foci are,

1. ~and~ (-1+\sqrt{10}, 2)\)

2. ~and~ (-1+\sqrt{8}, 2)\)

3. ~and~ (-1, 2+\sqrt{8})\)

4. ~and~ (-1, 2+\sqrt{10})\)

Q. The principal solution of a trigonometric equation lies in the interval

1. -π2, π2
2. $(-\pi ,\pi )$
3. $[0,2\pi )$

Q.

Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular x-axis.

(i) $x-\sqrt{3}y+8=0$, (ii) y - 2 = 0, (iii) x - y = 4

Q. If $0\le x<\frac{\pi }{2}$, then the number of values of $x$ for which $\sin x-\sin 2x+\sin 3x=0$, is :

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

Q.

If (cos$\theta$+isin$\theta$)(cos2$\theta$+isin2$\theta$)

(cosn$\theta$+isinn$\theta$)=1, then the value of $\theta$ is

1. 

2. 

3. 

4. 

Q.

If the equation 2 cos x + cos 2kx = 3 has only one solution then k is

1. 1

2. Integers

3. An irrational number

4. A rational number

Q.

The number of real roots of the equation $e^{sinx}-e^{-sinx}-4=0$ are

1. 1

2. 2

3. 0

4. Infinite

Q. If ${\log }_3{\left(\frac{2\sin 2x}{3}\right)}^2+1=0,x\in [0,2\pi ],$
then which among the following value(s) of $x$ satisfying the above equation

1. $x=\frac{\pi }{6}$
2. $x=\frac{4\pi }{3}$
3. $x=\frac{\pi }{3}$
4. $x=\frac{7\pi }{6}$

Q. 4.cosec x=-2

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. The set of all real values of $x$ for which the function $f\left(x\right)=\sqrt{\sin x+\cos x}+\sqrt{7x-x^2-6}$ takes real values is

1. $[1,6]-\{\frac{3\pi }{4},\frac{7\pi }{4}\}$
2. $[1,6]$
3. $[1,\frac{3\pi }{4}]\cup [\frac{7\pi }{4},6]$
4. $[1,\frac{\pi }{4}]\cup [\frac{7\pi }{4},6]$

Q.

Let $S\subset (0,\pi )$ denote the set of values of x satisfying the equation $8^{1+|cosx|+co{s}^{2}x+|co{s}^{3}x|+...to\infty }=4^3$ .Then, S=

1. 

2. 

3. 

4. 

Q. If $x\in [0,2\pi ]$, then the number of solutions of the equation ${81}^{{\cos }^{2}x}+{81}^{{\sin }^{2}x}=30$ is

Q. 3. cot x--V3

Q. If $x-y=\frac{1}{3}$ and ${\cos }^2\left(\pi x\right)-{\sin }^2\left(\pi y\right)=\frac{1}{2}$, then $(x,y)$ can be

1. $(\frac{2}{3},\frac{1}{3})$
2. $(\frac{13}{6},\frac{11}{6})$
3. $(\frac{7}{6},\frac{5}{6})$
4. $(\frac{19}{6},\frac{17}{6})$

Q. If $a,b\in [0,\pi ]$ and the equation $x^2+4+3\sin (ax+b)-2x=0$ has at least one solution, then the value of $(a+b)$ can be:

1. $\frac{7\pi }{2}$
2. $\frac{3\pi }{2}$
3. $\frac{9\pi }{2}$
4. $noneofthese$

Q.

Prove that

(i) $\frac{\text{sin}2x}{1-\text{cos}2x}=\text{cot}x$

(ii) $\frac{1-\text{cos2x}}{1+\text{cos}2x}={\text{tan}}^{2}x$

Q. $\cos x=-\frac{1}{3}$, x in quadrant III. Find the value of $\sin \frac{x}{2},\cos \frac{x}{2},\tan \frac{x}{2}$

Q. If $0\le x<2\pi$ , then the number of real values of x, which satisfy the equation $\cos x+\cos 2x+\cos 3x+\cos 4x=0$, is

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

Q.

Let $\theta \in (0,\frac{\pi }{4})and{t}_1=(tan\theta {)}^{tan\theta },t_2(tan\theta {)}^{cot}{t}_3=\left(cot\theta {)}^{tan\theta }and{t}_4\right(cot\theta {)}^{tan\theta }$ , then

1. 

2. 

3. 

4. 

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 combined equation of three sides of triangle(x2-y2)(2x+3y-6)=0 . if (sectheta, 1) is an interior point of the triangle then find value of 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. Let $f\left(x\right)=\int \frac{\sqrt{x}}{(1+x{)}^2}dx;(x\ge 0).$ Then $f\left(3\right)-f\left(1\right)$ is equal to

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.

If (cos$\theta$+isin$\theta$)(cos2$\theta$+isin2$\theta$)

(cosn$\theta$+isinn$\theta$)=1, then the value of $\theta$ is

1. 

2. 

3. 

4. 

Q. The number of solutions of the equation $\cos x=\sqrt{1-\sin 2x}$ where $x\in [0,2\pi ]$ is

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

Q. For $0<\theta <\frac{\pi }{2},$ the solution (s) of
${\sum }_{m=1}^{6}cosec(\theta +\frac{(m-1)\pi }{4})cosec(\theta +\frac{m\pi }{4})=4\sqrt{2}$ is/are

1. 
2. 
3. 
4.