Question

The general solution of the equation sin x -3 sin 2x + sin 3x =cos x -3 cos 2x + cos 3x is $(n\in Z)$

• A. $n\pi +\frac{\pi }{4}$
  
• B. $\frac{n\pi }{2}+\frac{\pi }{8}$
  
• C. $(-1{)}^n\frac{n\pi }{2}+\frac{\pi }{8}$
  
• D. $2n\pi +co{s}^{-1}\frac{2}{3}$

Answer 1

The correct option is B $\frac{n\pi }{2}+\frac{\pi }{8}$

The given equation is
sinx-3 sin 2x+ sin 3x =cos x -3 cos 2x + cos 3x
or 2 sin 2x cos x-3 sin 2x =2 cos2x cos x-3 cos 2x
or sin2x (2 cosx -3)=cos 2x (2 cos x-3)
or sin 2x =cos 2x (as $cosx\ne \frac{3}{2})$
or $tan2x=1\Rightarrow 2x=n\pi +\frac{\pi }{4}$
$\Rightarrow x=\frac{n\pi }{2}+\frac{\pi }{8},n\in Z$