General Solution of Trigonometric Equation

Q.

Count the number of triangles in given figure.

Q.

How do you simplify $1-{\cos }^2\theta$?

Q.

The value of ${\sin }^{-1}\left[\sin \left(10\right)\right]$ is

1. $10$

2. $10-3\mathrm{\pi }$

3. $3\mathrm{\pi }-10$

4. None of these.

Q.

$(\sin 4\mathrm{\theta })$ can be written as:

1. $4\sin \theta \left(1–2{\sin }^2\theta \right)\left(\sqrt{1–{\sin }^2\theta }\right)$

2. $2\sin \theta \cos \theta {\sin }^2\theta$

3. $4\sin \theta –6{\sin }^3\theta$

4. None of these

Q.

$\cos 2x+7=a(2-\sin x)$ can have a real solution for

1. All values of $a$

2. $a\in \left[2,6\right]$

3. $a\in \left(-\infty ,2\right)$

4. $a\in \left(0,\infty \right)$

Q.

Evaluate the following : $(1+\tan A+\sec A)(1+\cot A-\mathrm{cosec}A).$

Q.

What is the value of $\tan \left(180-\theta \right)$?

Q.

If $A+B+C=180°$, $\sum \tan \frac{A}{2}\tan \frac{B}{2}$ is

1. $0$

2. $1$

3. $2$

4. $3$

5. $4$

Q. The general solution of $\cos x+\sin x=\cos 2x+\sin 2x$ is

1. $\left\{2n\pi \right\}\cup \left\{\right(2n+1\left)\frac{\pi }{6}\right\},n\in Z$
2. $\left\{\right(2n+1\left)\frac{\pi }{2}\right\}\cap \left\{\right(2n+1\left)\frac{\pi }{6}\right\},n\in Z$
3. $\left\{\right(2n+1\left)\frac{\pi }{2}\right\}\cup \left\{\right(4n+1\left)\frac{\pi }{6}\right\},n\in Z$
4. $\left\{2n\pi \right\}\cup \left\{\right(4n+1\left)\frac{\pi }{6}\right\},n\in Z$

Q.

If $\cos 7\theta =\cos \theta -\sin 4\theta$, then the general value of $\theta$ is

1. $n\pi /4,n\pi /4+\pi /18$

2. $n\pi /3,n\pi /3+{(-1)}^n\pi /18$

3. $n\pi /4,n\pi /3+{(-1)}^n\pi /18$

4. $n\pi /6,n\pi /3+{(-1)}^n\pi /18$

Q. The equation $y=\sin x\sin (x+2)-{\sin }^2(x+1)$ represents a straight line lying in :

1. first, second and fourth quadrants
2. second and third quadrants only
3. first, third and fourth quadrants
4. third and fourth quadrants only

Q.

Why is $\cos \left(90°-x\right)=\sin \left(x\right)$ and $\sin \left(90°-x\right)=\cos \left(x\right)$?

Q.

In any $∆ABC$, $\frac{\tan {\displaystyle \frac{A}{2}}-\tan {\displaystyle \frac{B}{2}}}{\tan {\displaystyle \frac{A}{2}}+\tan {\displaystyle \frac{B}{2}}}$ is equal to:

1. $\frac{a-b}{a+b}$

2. $\frac{a-b}{c}$

3. $\frac{c}{a-b}$

4. None of these.

Q.

What is the value of $\cos \left(\frac{\mathrm{\pi }}{8}\right)$?

Q. The principal solution(s) for $\cos x=-\frac{1}{\sqrt{2}}$ is/are

1. $\frac{3\pi }{4}$
2. $\frac{5\pi }{4}$
3. $\frac{11\pi }{4}$
4. $\frac{13\pi }{4}$

Q. The principal solution(s) for $\tan x=-1$ is/are

1. $x=\frac{3\pi }{4}$
2. $x=\frac{11\pi }{4}$
3. $x=\frac{7\pi }{4}$
4. $x=-\frac{\pi }{4}$

Q.

If ${\sin }^2\left(\theta \right)=\frac{1}{4}$, then the most general value of $\left(\theta \right)$ is

1. $2n\mathrm{\pi }\pm {\left(-1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{6}$

2. $\left(\frac{n\mathrm{\pi }}{2}\right)\pm {\left(-1\right)}^n\frac{\mathrm{\pi }}{6}$

3. $n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6}$

4. $2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6}$

Q.

$\text{The total number of solutions of the equation}\cos x.\cos 2x.\cos 3x=\frac{1}{4}\text{in}[0,\pi ]\text{is}$

1. $7$

2. $6$

3. $4$

4. $2$

Q. The number of solutions of the equation $1+{\sin }^{4}x={\cos }^{2}3x,x\in [-\frac{5\pi }{2},\frac{5\pi }{2}]$ is :

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

Q.

How do you find the derivative of $y=\tan \left(x\right)$ using first principle ?

Q. Number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in $x\in [0,2\pi ]$ is

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

Q.

The most general solution of $\sqrt{3}\cos \theta +\sin \theta =\sqrt{2}$ is

1. $\theta =n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{6}$

2. $\theta =n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{6}$

3. $\theta =2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{6}$

4. $\theta =2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{6}$

Q.

If $\sec \left(\theta \right)+\tan \left(\theta \right)=k$, then $\cos \left(\theta \right)$ equals

1. $\frac{k^2+1}{2k}$

2. $\frac{2k}{k^2+1}$

3. $\frac{k}{k^2+1}$

4. $\frac{k}{k^2-1}$

Q.

If $y={\cos }^{2}x+se{c}^{2}x$, then

1. $y\le 2$

2. $y\le 1$

3. $y\ge 2$

4. $1<y<2$

Q. The set of values of $x$ for which $\frac{\tan 3x-\tan 2x}{1+\tan 3x\cdot \tan 2x}=1$ is (where $n\in Z$)

1. $\{n\pi +\frac{\pi }{4}\}$
2. $\{n\pi +\frac{\pi }{6}\}$
3. $\{n\pi +\frac{\pi }{3}\}$
4. $\varphi$

Q.

$\sin 47°+\sin 61°-\sin 11°-\sin 25°$ is equal to:

1. $\sin 7°$

2. $\cos 7°$

3. $\sin 36°$

4. $\cos 36°$

Q. The number of ordered pairs $(x,y)$ satisfying $|x|+|y|=3$ and $\sin \left(\frac{\pi x^2}{3}\right)=1$ is less than equal to

1. $7$
2. $8$
3. $9$
4. $10$

Q. The number of distinct solutions of the equation,
${\log }_{\frac{1}{2}}|\sin x|=2-{\log }_{\frac{1}{2}}|\cos x|$ $\text{in the interval}[0,2\pi ],\text{is}$

Q. Find the value of : cos75.sin75

Q.

$\text{The number of values of x in the interval}[0,5\pi ]satisfyingtheequation3si{n}^{2}x-7sinx+2=0is$

1. 5

2. 0

3. 6

4. 10