Answer

Verified

373.2k+ views

**Hint:**First simplify double angle i.e. $2x$ in terms of single angle i.e. $x$ using the double angle formula for sine function which is given as follows:

$ 2\sin x\cos x$. Then further simplify and find the principal solution and then the general solution.

**Complete step by step solution:**

To solve the given trigonometric expression $\sin \left( {2x} \right) - \sin \left( x \right) = 0$, we will first convert the double angle $(2x)$ into single angle $(x)$ with the

help of double angle formula for sine function which is given as

$ 2\sin x\cos x$

Therefore the given trigonometric equation will be written as

$

\Rightarrow \sin\left( {2x} \right) - \sin \left( x \right) = 0 \\

\Rightarrow 2\sin x\cos x - \sin x = 0 \\

$

Now taking $\sin x$ common in left hand side, we will get

$

\Rightarrow 2\sin x\cos x - \sin x = 0 \\

\Rightarrow \sin x\left( {2\cos x - 1} \right) = 0 \\

$

Now from the final equation, that is $\sin x\left( {2\cos x - 1} \right) = 0$ two possibilities are creating here for the solution of the given trigonometric equation,

$

\Rightarrow \sin x = 0\;{\text{or}}\;\left( {2\cos x - 1} \right) = 0 \\

\Rightarrow \sin x = 0\;{\text{or}}\;\cos x = \dfrac{1}{2} \\

$

We know that at $x = 0\;{\text{and}}\;x = \dfrac{\pi }{3}$ sine and cosine function have their respective values of $0\;{\text{and}}\;\dfrac{1}{2}$

Therefore the required solution for the trigonometric equation $\sin \left( {2x} \right) - \sin \left( x \right) = 0$ is given by

$x = 0\;{\text{and}}\;x = \dfrac{\pi }{3}$

Hence we will now write its general solution,

We know that the general solution of $\sin x = 0$ is given as follows

$x = \pm n\pi ,\;{\text{where}}\;n \in I$

And general solution of $\cos x = \dfrac{1}{2}$ is given as

$x = 2n\pi \pm \dfrac{\pi }{3},\;{\text{where}}\;n \in I$

**Therefore the general solution of the given trigonometric equation $\sin \left( {2x} \right) - \sin \left( x \right) = 0$ is given as**

$x = \left( { \pm n\pi \cup 2n\pi \pm \dfrac{\pi }{3}} \right),\;{\text{where}}\;n \in I$

$x = \left( { \pm n\pi \cup 2n\pi \pm \dfrac{\pi }{3}} \right),\;{\text{where}}\;n \in I$

**Note:**The double angle formula is a special case of the addition formula of sine angle in which we consider both the arguments equal. Normally periodic functions have three types of solution that are principal solution which is the smallest possible solution, particular solution which lies according to the conditions given in the problem and general solution which is set of each and every possible solution.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE