Question

The number of solution of the equation $ \sin x+\sin 5x=\sin 3x $ tying in the interval $ [0,\,\pi ] $ is

• A. $ 4 $
• B. $ 6 $
• C. $ 5 $
• D. $ 2 $

Answer 1

Answer: (B)

$ \sin x+\sin 5x=\sin 3x $
$ \Rightarrow $ $ 2\sin \left( \frac{x+5x}{2} \right).\cos \left( \frac{5x-x}{2} \right)=\sin 3x $
$ \Rightarrow $ $ 2\sin 3x.cos\,2x=sin3x $
$ \Rightarrow $ $ \sin 3x\,(2\cos \,2x-1)=0 $
$ \Rightarrow $ $ \sin 3x=0 $ and $ 2\cos 2x-1=0 $
$ \Rightarrow $ $ \sin 3x=sin0 $ and $ \cos 2x=\frac{1}{2}=\cos \frac{\pi }{3} $
$ \Rightarrow $ $ 3x=n\pi $ and $ 2x=2n\pi \pm \frac{\pi }{3} $
$ \Rightarrow $ $ x=\frac{n\pi }{3} $ and $ x=n\pi \pm \frac{\pi }{6} $
where $ n\in z, $
$ \Rightarrow $ $ x=0,\,\frac{\pi }{3},\frac{2\pi }{3},\frac{3\pi }{3} $
and $ x=\frac{\pi }{6},\frac{5\pi }{6} $
For $ n=0,1,2,3.... $
So, the total number of solution lie between
$ [0,\,\pi ] $ $ x=0,\frac{\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{5\pi }{6},\frac{3\pi }{3} $