Question

The solutions of $x+\sin \,5x=\sin 3x$ in $\left( 0,\frac{\pi }{2} \right)$ are

• A. $\frac{\pi }{4},\frac{\pi }{10}$
• B. $\frac{\pi }{6},\frac{\pi }{3}$
• C. $\frac{\pi }{4},\frac{\pi }{2}$
• D. $\frac{\pi }{8},\frac{\pi }{16}$

Answer 1

Answer: (B)

$\frac{\pi }{6},\frac{\pi }{3}$

Answer 2

$\sin \,\,x+\,\sin \,5x=\,\sin \,3x$ $\Rightarrow$ $2\,\,\sin \,3x\,cos\,2x\,=\sin \,3x$ $\Rightarrow$ $\sin 3x(2\cos 2x-1)=0$ $\Rightarrow$ $\sin \,\,3x=0$ or $2\,\cos \,2x-1=0$ if $\sin \,\,3x=0$ $\Rightarrow$ $3x=0,$ or $\pi$ $\Rightarrow$ $x=0$ or $x=\frac{\pi }{3}$ And if $2\,\,\cos \,2x-1=0$ $\Rightarrow$ $\cos \,\,2x=\frac{1}{2}\,=\,\cos \,\frac{\pi }{3}$ $\Rightarrow$ $2x=\frac{\pi }{3}\Rightarrow x=\frac{\pi }{6}$ $\therefore$ Solutions in $\left( 0,\frac{\pi }{2} \right)$ are $\frac{\pi }{3},\frac{\pi }{6}.$