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# General Solution of Trigonometric Equations - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Trigonometric Equations, General Solution of some Standard Equations (Part 1) is considered one of the most asked concept.

• 51 Questions around this concept.

## Solve by difficulty

If $\dpi{100} 5\left ( \tan ^{2}x-\cos ^{2}x \right )=2\cos \: 2x+9$

then the value of $\dpi{100} \cos 4x$ is :

In a $\Delta PQR$, if  $\dpi{100} 3\sin P+4\cos Q=6\:\: and\: \: 4\sin Q+3\cos P=1$, then the angle R is equal to

The general solution of the equation

$\sum_{r=1}^n \cos \left(r^2 x\right) \sin (r x)=\frac{1}{2}$  is

Let n be a fixed positive integer such that $\sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\frac{\sqrt{n}}{2},$  then

The number of solutions of the equation $\sin x+2 \sin 2 x=3+\sin 3 x$ in the interval $[0, \pi]$ is:

Let $\mathrm{f(\theta)=3\left(\sin ^4\left(\frac{3 \pi}{2}-\theta\right)+\sin ^4(3 \pi+\theta)\right)-2\left(1-\sin ^2 2 \theta\right) \quad and \quad S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=\right.\left.-\frac{\sqrt{3}}{2}\right\}. If 4 \beta=\sum_{\theta \in S} \theta, then f(\beta)}$is equal to

Let $S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}$ and  $\mathrm{b}=\sum_{\mathrm{x} \in \mathrm{S}} \tan ^2\left(\frac{\mathrm{x}}{3}\right), then \frac{1}{6}(\beta-14)^2$is equal to:

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The most general solution of $\cot \theta = - \sqrt 3 \: \: and \: \: \csc \theta = -2$is :

If $tan\; x=-\frac{4}{7}$ , $x$  lies in the second quadrant. Then the value of  $cot\: x$ is

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$If 2 \tan ^2 \theta-5 \sec \theta=1$ has exactly 7 solutions in the interval $[0, n \pi / 2]$, for the least value of n $\in N, then \sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}$   is Equal to.

## Concepts Covered - 4

Trigonometric Equations

Trigonometric Equations

Trigonometric equations are, as the name implies, equations that involve trigonometric functions.

Solution of Trigonometric Equation

The value of an unknown angle which satisfies the given trigonometric equation is called a solution or root of the equation. For example, 2sinӨ = 1, clearly Ө = 300 satisfies the equation; therefore, 30is a solution of the equation. Now trigonometric equation ususally has infinite solutions due to periodic nature of trigonometric functions. So this equation also has (360+30)o,(720+30)o,(-360+30)o and so on, as its solutions.

Principal Solution

The solutions of a trigonometric equation that lie in the interval [0, 2π). For example, 2sinӨ = 1 , then the two values of sinӨ between 0 and 2π are  π/6 and 5π/6. Thus, π/6 and 5π/6 are the principal solutions of equation 2sinӨ = 1.

General Solution

As trigonometric functions are periodic, solutions are repeated within each period, so, trigonometric equations may have an infinite number of solutions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.

Some Important General Solutions of Equations

 $\mathbf{Equations}$ $\mathbf{Solution}$ $\sin\theta=0$ $\theta=n \pi, \quad n \in \mathbb{I}$ $\cos\theta=0$ $\theta=(2 n+1) \frac{\pi}{2}, \quad n \in \mathbb{I}$ $\tan\theta=0$ $\theta=n \pi, \quad n \in \mathbb{I}$ $\sin\theta=1$ $\theta=(4 n+1) \frac{\pi}{2}, \quad n \in \mathbb{I}$ $\cos\theta=1$ $\theta=2 n \pi, \quad n \in \mathbb{I}$ $\sin\theta=-1$ ${\theta}=(4 n-1) \frac{\pi}{2}, \quad n \in \mathbb{I}$ $\cos\theta=-1$ $\theta=(2 n+1) \pi, \quad n \in \mathbb{I}$ $\cot\theta=0$ $\theta=(2 n+1) \frac{\pi}{2}, \quad n \in \mathbb{I}$
General Solution of some Standard Equations (Part 1)

General Solution of some Standard Equations (Part 1)

1. sin Ө = sin α

$\\\text{Given,}\;\sin\theta=\sin\alpha\;\;\Rightarrow \sin\theta-\sin\alpha=0\\\\\mathrm{\Rightarrow 2\cos\frac{\theta+\alpha}{2}\sin\frac{\theta-\alpha}{2}=0}\\\\\mathrm{\Rightarrow \cos\frac{\theta+\alpha}{2}=0\;\;\;\;or\;\;\;\;\sin\frac{\theta-\alpha}{2}=0}\\\\\mathrm{\Rightarrow\frac{\theta+\alpha}{2}=(2n+1)\frac{\pi}{2}\;\;\;\;or\;\;\;\; \frac{\theta-\alpha}{2}=n\pi,\;\;\;n\in\mathbb{I}}\\\\\mathrm{\Rightarrow \theta=(2n+1)\pi-\alpha\;\;\;\;or\;\;\;\;\theta=2n\pi+\alpha,\;\;\;n\in\mathbb{I}}\\\mathrm{\Rightarrow \theta=(any\;odd\;multiple\;of\;\pi)-\alpha\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(i)}\\\mathrm{\Rightarrow \theta=(any\;even\;multiple\;of\;\pi)+\alpha\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(ii)}\\\mathrm{from\;(i)\;and\;(ii)}\\\mathrm{\theta=n\pi+(-1)^n\alpha,\;\;n\in\mathbb{I}}$

2. cos Ө = cos α

$\\\mathrm{\Rightarrow \cos \alpha-\cos \theta=0}\\\mathrm{\Rightarrow 2 \sin \frac{\alpha+\theta}{2} \sin \frac{\theta-\alpha}{2}=0}\\\mathrm{\Rightarrow \quad \sin \frac{\alpha+\theta}{2}=0\;\;\;or\;\;\;\sin \frac{\theta-\alpha}{2}=0}\\\mathrm{\Rightarrow \quad \frac{\alpha+\theta}{2}=n \pi \text { or } \frac{\theta-\alpha}{2}=n \pi, n \in \mathbb{I}}\\\mathrm{\Rightarrow \quad \theta=2 n \pi-\alpha \text { or } \theta=2 n \pi+\alpha, n \in Z}\\\mathrm{\Rightarrow \quad \theta= 2 n \pi \pm \alpha, n \in \mathbb{I}}$

3. tan Ө = tan α

$\\\text { Given, } \tan \theta=\tan \alpha\\\mathrm{\Rightarrow \quad \frac{\sin \theta}{\cos \theta}=\frac{\sin \alpha}{\cos \alpha}}\\\mathrm{\Rightarrow \quad \sin \theta \cos \alpha-\cos \theta \sin \alpha=0}\\\mathrm{\Rightarrow \quad \sin (\theta-\alpha)=0}\\\mathrm{\Rightarrow \quad\theta-\alpha=n \pi}\\\mathrm{\Rightarrow \quad \theta=n \pi+\alpha, \text { where } n \in \mathbb{I}}$

General Solution of some Standard Equations (Part 2)

General Solution of some Standard Equations (Part 2)

4. sin2 Ө = sin2 α

$\begin{array}{l}{\Rightarrow \quad \sin ^{2} \theta=\sin ^{2} \dot{\alpha}} \\ {\Rightarrow \quad \sin (\theta+\alpha) \sin (\theta-\alpha)=0} \\\mathrm{\because we\;are\;using\;the\;identity,\sin (A+B) \sin (A-B)=\sin ^{2} A-\sin ^{2} B}\\ {\Rightarrow \sin (\theta+\alpha)=0 \text { or } \sin (\theta-\alpha)=0} \\ {\Rightarrow \theta+\alpha=n \pi \text { or } \theta-\alpha=n \pi, n \in \mathbb{I}} \\ {\Rightarrow \theta=n \pi \pm \alpha \in \mathbb{I}}\end{array}$

Note:

The general solution of the equation cos2 Ө = cos2 α and tan2 Ө = tan2 α is also  $\mathrm{{ \theta=n \pi \pm \alpha \in \mathbb{I}}} .$

Important Points to remember while solving trigonometric equations

Important Points to remember while solving trigonometric equations

1. While solving a trigonometric equation, squaring the equation at any step should be avoided as much as possible. If squaring is necessary, check the solution for values that do not satisfy the original equation.

2. Never cancel terms containing unknown terms on the two sides which are in product. It may cause the loss of a genuine solution.

3. The answer should not contain such values of angles which make any of the terms undefined or infinite.

4. Domain should not change while simplifying the equation. If it changes, necessary corrections must be made.

5. Check that the denominator is not zero at any stage while solving the equations.

Example:

Solve sin x + cos x = 1

Solution:

$\\\mathrm{Given \;equation\;is,\;\sin x+\cos x=1}\\\mathrm{If\;we\;square\;both\;side,}\\\mathrm{\left (\sin x+\cos x \right )^2=1^2}\\\mathrm{\Rightarrow \quad \sin^2x+\2\sin x\cos x+\cos^2 x=1}\\\mathrm{\Rightarrow \quad\sin2x=0}\\\mathrm{\Rightarrow \quad2x=n\pi,\;\;n\in\mathbb{I}}\\\mathrm{\Rightarrow \quad x=\left (n\pi \right )/2,\;\;n\in\mathbb{I}}$
$\begin{array}{l}{\text { But for } n=2,6,10, \ldots} \\ {\sin x+\cos +=-1 \text { which contradicts the given equation. }}\end{array}$

$\begin{array}{l}{\text { Also for } n=3,7,11, \ldots} \\ {\sin x+\cos x=-1}\end{array}$

Hence only n = 0, 4, 8, 12, ..... and n = 1, 5, 9, .... satisfy the equation

$\text { Hence, the solution is } x=4n\pi/2 = 2 n \pi \text { or } x=(4 n+1) \frac{\pi}{2}$

## Study it with Videos

Trigonometric Equations
General Solution of some Standard Equations (Part 1)
General Solution of some Standard Equations (Part 2)

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## Books

### Reference Books

#### Trigonometric Equations

Mathematics for Joint Entrance Examination JEE (Advanced) : Trigonometry

Page No. : 4.1

Line : 1

#### General Solution of some Standard Equations (Part 1)

Mathematics for Joint Entrance Examination JEE (Advanced) : Trigonometry

Page No. : 4.4

Line : 42

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