General Solution of Trigonometric Equation

Q.

Prove the following:

$\text{cos}(\frac{\pi }{4}-x)\text{cos}(\frac{\pi }{4}-y)$$\text{-}sin(\frac{\pi }{4}-x)\text{sin}(\frac{\pi }{4}-y)$ = sin (x+y)

Q. The solution set of $\sec x=\sec (\pi +x)$ is

1. $x=n\pi ,n\in Z$
2. $x=2n\pi \pm \frac{\pi }{2},n\in Z$
3. $x=(2n+1)\frac{\pi }{2},n\in Z$
4. $x=\varphi$

Q. 6. Find the derivatives of the functions w.r.t. x from first principles: i) e*x or e to the power x ii) e*sinx or e to the power sinx

Q.

If $tan{1}^{\circ }tan{2}^{\circ }.............tan{89}^{\circ }=x^2-8$, then the value of x can be

1. ±4

2. ±2

3. ±3

4. ±1

Q.

The number of values of y in $[-2\pi ,2\pi ]$ satisfying the equation |sin 2x| = |cos 2x| = |sin y| is

1. 3

2. 4

3. 1

4. 2

Q.

If for a $△ABC,cotA.cotB.cotC>0$ then the triangle is

1. All of these

2. Right angled

3. Acute angled

4. Obtuse angled

Q. Which of the following set of values of $x$ satisfies the equation $2^{2{\sin }^{2}x-3\sin x+1}+2^{2-2{\sin }^{2}x+3\sin x}=9$
(where $n\in Z$)

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

Q. The number of solutions of the equation $e^{\sin x}-e^{-\sin x}-4=0,(e\simeq 2.73)$ is

1. $1$
2. $2$
3. $3$
4. no solution

Q.

If ${\log }_{|sinx|}|cosx|+{\log }_{|cosx|}|sinx|=2$ then |tanx|

1. 0

2. 2

3. 3

4. 1

Q. The general solution(s) of $4\sin \theta \sin 2\theta \sin 4\theta =\sin 3\theta$ can be

1. $\frac{n\pi }{2},n\in Z$
2. $n\pi ,n\in Z$
3. $(3n\pm 1)\frac{\pi }{3},n\in Z$
4. $(3n\pm 1)\frac{\pi }{9},n\in Z$

Q. The set of values of $x$ for which $\frac{\tan 3x-\tan 2x}{1+\tan 3x\tan 2x}=1$ is

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

Q.

The complete solution set of the inequality $[co{t}^{-1}x{]}^2-6[co{t}^{-1}x]+9\le 0,$ where $[.]$ denotes greatest integer function is

1. $(-\infty ,cot2)$
2. $(-\infty ,cot3)$
3. $(cot2,cot3)$
4. $(cot2,\infty )$

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. The solution set for the inequality $\sin \alpha {\cos }^3\alpha >{\sin }^3\alpha \cos \alpha$ where $\alpha \in (0,\pi )$ is

1. $(0,\frac{\pi }{4})\cup (\frac{\pi }{2},\frac{3\pi }{4})$
2. $(0,\frac{\pi }{2})\cup (\frac{\pi }{2},\frac{3\pi }{4})$
3. $(\frac{\pi }{4},\frac{\pi }{2})\cup (\frac{\pi }{2},\frac{3\pi }{4})$
4. $(0,\frac{\pi }{4})\cup (\frac{3\pi }{4},\pi )$

Q. The value(s) of $x$ satisfying both the equations $\sin x=-\frac{1}{2}$ and $\tan x=\frac{1}{\sqrt{3}}$ in $[0,2\pi ]$ simultaneously is :

1. $\frac{\pi }{6}$
2. $\frac{7\pi }{6}$
3. $\frac{11\pi }{6}$
4. $\frac{7\pi }{6},\frac{11\pi }{6}$

Q. The solution set for $2{\cos }^2\theta +\sin \theta \le 2,$where $\frac{\pi }{2}\le \theta \le \frac{3\pi }{2}$, is

1. $[\frac{\pi }{2},\frac{5\pi }{6}]\cup [\pi ,\frac{3\pi }{2}]$
2. $[\frac{\pi }{2},\frac{2\pi }{3}]\cup [\pi ,\frac{3\pi }{2}]$
3. $[\frac{2\pi }{3},\frac{5\pi }{6}]\cup [\pi ,\frac{3\pi }{2}]$
4. $[\pi ,\frac{3\pi }{2}]$

Q. $$\begin{array}{cc}TrigonometricEquations& GeneralSolutions\\ 1.7co{s}^{2}x+si{n}^{2}x=4& P.n\pi \pm \frac{\pi }{4},wheren\in I\\ 2.si{n}^{2}x=\frac{1}{2}& Q.n\pi \pm \frac{\pi }{3},wheren\in I\\ 3.ta{n}^{2}x+3=0& R.n\pi \pm \frac{\pi }{6},wheren\in I\\ 4.3ta{n}^{2}x-1=0& S.Nosolution\end{array}$$

1. 1 - Q, 2 - P, 3 - S, 4 - R

2. 1 - Q, 2 - P, 3 - Q, 4 - R

3. 1 - R, 2 - R, 3 - Q, 4 - R

4. 1 - R, 2 - P, 3 - S, 4 - P

Q. If $\tan a,\tan b,\tan c,\tan d$ are the roots of the eqn $\tan (45°+\theta )=3\tan 3\theta$, then
$\frac{1}{\tan a}+\frac{1}{\tan b}+\frac{1}{\tan c}+\frac{1}{\tan d}$ is

Q. If $e^{|\sin x|}+e^{-|\sin x|}+4a=0$ will have exactly four different solutions in $[0,2\pi ]$, then

1. $a\in R$
2. $a\in [-\frac{e}{4},-\frac{1}{4}]$
3. $a\in [\frac{-1-e^2}{4e},\infty ]$
4. no real value of $a$ exists

Q. The solution set of the equation $\sin 3x+\cos 2x=-2$ is
(where $n\in Z$)

1. $\frac{(2n+1)\pi }{2}$
2. $\frac{(4n-1)\pi }{2}$
3. $\frac{(2n-1)\pi }{2}$
4. $\frac{(4n+1)\pi }{2}$

Q. The set of all $x$ in $(-\pi ,\pi )$ satisying $|4\cos x-1|<\sqrt{5}$ is

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

Q. General solution of the equation $\tan x+\tan 2x+\sqrt{3}\tan x\tan 2x=\sqrt{3}$ is :

1. $\frac{n\pi }{3}+\frac{\pi }{9},n\in Z$
2. $n\pi +\frac{\pi }{9},n\in Z$
3. $n\pi -\frac{\pi }{9},n\in Z$
4. $\frac{n\pi }{2}+\frac{\pi }{3},n\in Z$

Q. If $\tan \left(\cot x\right)=\cot \left(\tan x\right)$, then the least positive value of $\cot x+\tan x$ is

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

Q. The general solution(s) of the equation $\sec 4\theta -\sec 2\theta =2$ can be

1. $(2n+1)\frac{\pi }{12},n\in Z$
2. $(2n+1)\frac{\pi }{10},n\in Z$
3. $(2n+1)\frac{\pi }{2},n\in Z$
4. $(2n+1)\frac{\pi }{4},n\in Z$

Q. cos 4 x = cos 2x·

Q. Find the number of points where $f\left(x\right)=[\sin x+\cos x]$ (where $\left[\right]$ denotes the greatest integer function), $x\in (0,2\pi )$

1. $5$
2. $7$
3. $10$
4. $6$

Q.

Point D, E are taken on the side BC of the triangle ABC, such that BD = DE = EC. If $\angle BAD=x,$ $\angle DAE=y,$ $\angle EAC=z,$ then the value of $\frac{sin(x+y)sin(y+z)}{sinxsinz}$

1. 2

2. 4

3. 3

4. 1

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 complete solution set of the inequality $[co{t}^{-1}x{]}^2-6[co{t}^{-1}x]+9\le 0,$ where $[.]$ denotes greatest integer function is

1. $(-\infty ,cot2)$

2. $(-\infty ,cot3)$

3. $(cot2,cot3)$

4. $(cot2,\infty )$

Q. General solution of the equation $\tan 5x=\cot 5x$ is :

1. $n\pi \pm \frac{\pi }{20},n\in Z$
2. $\frac{n\pi }{5}+\frac{\pi }{20},n\in Z$
3. $\frac{n\pi }{5}\pm \frac{\pi }{4},n\in Z$
4. $\frac{n\pi }{5}\pm \frac{\pi }{20},n\in Z$