Question

The possible value of $\theta \in (0,\pi )$ such that $\sin \left(\theta \right)+\sin \left(4\theta \right)+\sin \left(7\theta \right)=0$ are : ?

• A. $\frac{2\pi }{9},\frac{\pi }{4},\frac{4\pi }{9},\frac{\pi }{2},\frac{3\pi }{4},\frac{8\pi }{9}$
• B. $\frac{\pi }{4},\frac{5\pi }{12},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{8\pi }{9}$
• C. $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{35\pi }{36}$
• D. $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{8\pi }{9}.$

Answer 1

The correct option is A $\frac{2\pi }{9},\frac{\pi }{4},\frac{4\pi }{9},\frac{\pi }{2},\frac{3\pi }{4},\frac{8\pi }{9}$

Here, given $\theta \in (0,\pi )$

and $\sin \left(\theta \right)+\sin \left(4\theta \right)+\sin \left(7\theta \right)=0\Rightarrow [\sin \left(7\theta \right)+\sin \left(\theta \right)]+\sin \left(4\theta \right)=0$

We know that,

$\sin C+\sin D=2\sin \left(\frac{C+D}{2}\right)\cos \left(\frac{C-D}{2}\right)\therefore [\sin \left(7\theta \right)+\sin \left(\theta \right)]+\sin \left(4\theta \right)=0\Rightarrow 2\sin \left(\frac{7\theta +\theta }{2}\right)\cos \left(\frac{7\theta -\theta }{2}\right)+\sin \left(4\theta \right)=0\Rightarrow 2\sin \left(4\theta \right)\cos \left(3\theta \right)+\sin \left(4\theta \right)=0\Rightarrow \sin \left(4\theta \right)[2\cos \left(3\theta \right)+1]=0\Rightarrow \sin \left(4\theta \right)=0,2\cos \left(3\theta \right)+1=0$

For,

$\sin \left(4\theta \right)=0\Rightarrow 4\theta ={\sin }^{-1}\left(0\right)\Rightarrow 4\theta =\pi ,2\pi ,3\pi ,4\pi ,...\Rightarrow \theta =\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi ,...$

Here, $\theta \in (0,\pi )$

And

$2\cos \left(3\theta \right)+1=0\Rightarrow \cos \left(3\theta \right)=-\frac{1}{2}\Rightarrow 3\theta ={\cos }^{-1}(-\frac{1}{2})\Rightarrow 3\theta =\pi -\frac{\pi }{3},\pi +\frac{\pi }{3},3\pi -\frac{\pi }{3},3\pi +\frac{\pi }{3},...\Rightarrow \theta =\frac{2\pi }{9},\frac{4\pi }{9},\frac{8\pi }{9},...$

Here, $\theta \in (0,\pi )$

$\therefore$ The possible values of $\theta =\dfrac { 2\pi }{ 9 } ,\dfrac { \pi }{ 4 } ,\dfrac { 4\pi }{ 9 } ,\dfrac { \pi }{ 2 } ,\dfrac { 3\pi }{ 4 } ,\dfrac { 8\pi }{ 9 } .$