Question

Solve the equality: $2\sin 11x+\cos 3x+\sqrt{3}\sin 3x=0$

Answer 1

$2\sin 11x+\cos 3x+\sqrt{3}\sin 3x=0$

$\Rightarrow \sin 11x+\frac{\sqrt{3}}{2}\sin 3x+\frac{1}{2}\cos 3x=0$

$\Rightarrow \sin 11x+\sin (3x+\frac{\pi }{6})=0$

$\Rightarrow 2\sin (7x+\frac{\pi }{12})\cos (4x-\frac{\pi }{12})=0$

$\therefore \sin (7x+\frac{\pi }{12})=0,\cos (4x-\frac{\pi }{12})=0$

$\therefore 7x+\frac{\pi }{12}=x\pi ,4x-\frac{\pi }{12}=(2x+1)\frac{\pi }{2}$

$\Rightarrow 7x=x\pi -\frac{\pi }{12},4x=(2x+1)\frac{\pi }{2}+\frac{\pi }{12}$

$x=\frac{x\pi }{7}-\frac{\pi }{84},\therefore x=(2x+1)\frac{\pi }{8}+\frac{\pi }{48}$