Answer
454.5k+ views
Hint: Use $\sin 2x = 2\sin x\cos x$ and $\sin 3x = 3\sin x - 4{\sin ^3}x$ and then simplify the equation.
According to question, the given equation is:
$\sin x + 2\sin 2x - \sin 3x = 3$
We know that, $\sin 2x = 2\sin x\cos x$ and $\sin 3x = 3\sin x - 4{\sin ^3}x$, using these two results, we’ll get:
$
\Rightarrow \sin x + 2(2\sin x\cos x) - (3\sin x - 4{\sin ^3}x) = 3 \\
\Rightarrow \sin x + 4\sin x\cos x - 3\sin x + 4{\sin ^3}x = 3 \\
\Rightarrow \sin x[1 + 4\cos x - 3 + 4{\sin ^2}x] = 3 \\
\Rightarrow \sin x[4\cos x + 4{\sin ^2}x - 2] = 3 \\
$
Now, putting ${\sin ^2}x = 1 - {\cos ^2}x$ we’ll get:
\[
\Rightarrow \sin x[4\cos x + 4 - 4{\cos ^2}x - 2] = 3 \\
\Rightarrow \sin x[2 - (4{\cos ^2}x - 4\cos x)] = 3 \\
\Rightarrow \sin x[2 - (4{\cos ^2}x - 4\cos x + 1) + 1] = 3 \\
\Rightarrow \sin x[3 - {(2\cos x - 1)^2}] = 3 \\
\]
We know that, \[3 - {(2\cos x - 1)^2} \geqslant 3\] Therefore, for the above equation to satisfy, we have:
$ \Rightarrow \sin x = 1$ and \[{(2\cos x - 1)^2} = 0\]
$ \Rightarrow x = \frac{\pi }{2}$ and $\cos x = \frac{1}{2}$
$ \Rightarrow x = \frac{\pi }{2}$ and $x = \frac{\pi }{3}$
But $x$ cannot have two values at the same time. Therefore, the above equation will not have any solution. And option (D) is correct.
Note: Both $\sin x = 1$ and $\cos x = \frac{1}{2}$ cannot satisfy at the same time. If in any equation, the value of $\sin x$ is coming out to be $1,$ for any value of $x,$ then the value of $\cos x$ must be $0$ for that particular $x.$
According to question, the given equation is:
$\sin x + 2\sin 2x - \sin 3x = 3$
We know that, $\sin 2x = 2\sin x\cos x$ and $\sin 3x = 3\sin x - 4{\sin ^3}x$, using these two results, we’ll get:
$
\Rightarrow \sin x + 2(2\sin x\cos x) - (3\sin x - 4{\sin ^3}x) = 3 \\
\Rightarrow \sin x + 4\sin x\cos x - 3\sin x + 4{\sin ^3}x = 3 \\
\Rightarrow \sin x[1 + 4\cos x - 3 + 4{\sin ^2}x] = 3 \\
\Rightarrow \sin x[4\cos x + 4{\sin ^2}x - 2] = 3 \\
$
Now, putting ${\sin ^2}x = 1 - {\cos ^2}x$ we’ll get:
\[
\Rightarrow \sin x[4\cos x + 4 - 4{\cos ^2}x - 2] = 3 \\
\Rightarrow \sin x[2 - (4{\cos ^2}x - 4\cos x)] = 3 \\
\Rightarrow \sin x[2 - (4{\cos ^2}x - 4\cos x + 1) + 1] = 3 \\
\Rightarrow \sin x[3 - {(2\cos x - 1)^2}] = 3 \\
\]
We know that, \[3 - {(2\cos x - 1)^2} \geqslant 3\] Therefore, for the above equation to satisfy, we have:
$ \Rightarrow \sin x = 1$ and \[{(2\cos x - 1)^2} = 0\]
$ \Rightarrow x = \frac{\pi }{2}$ and $\cos x = \frac{1}{2}$
$ \Rightarrow x = \frac{\pi }{2}$ and $x = \frac{\pi }{3}$
But $x$ cannot have two values at the same time. Therefore, the above equation will not have any solution. And option (D) is correct.
Note: Both $\sin x = 1$ and $\cos x = \frac{1}{2}$ cannot satisfy at the same time. If in any equation, the value of $\sin x$ is coming out to be $1,$ for any value of $x,$ then the value of $\cos x$ must be $0$ for that particular $x.$
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)