Answer

Verified

415.8k+ 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

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The states of India which do not have an International class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

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

Why is there a time difference of about 5 hours between class 10 social science CBSE

Name the three parallel ranges of the Himalayas Describe class 9 social science CBSE