Question

Sin x + sin 3x = sin 4x

Answer 1

$sinx+sin3x=sin4x \ sinx+sin3x=2sin2x*cos2x \ 2sin frac{x+3x}{2} *cos frac{x-3x}{2}=2sinxcosx*(1-2 sin^{2}x) \ 2sin2x*cos(-x)-sin2x(1-2cos^{2} x-1)=0 \ sin2x(2cos(-x)+2cos^{2}x+1-1)=0 \ 1) sin2x=0 \ 2x= pi n,n e Z \ x= frac{ pi }{2}n, neZ \ ******* \ 2cos(-x)+cos2x+1=0 \ 4cos frac{-x+2x}{2}*cos frac{-x-2x}{2}-1=0 \ 4cos frac{x}{2}*cos frac{-3x}{2}=1 \ 2) 4cos frac{x}{2}=1 \ cos frac{x}{2}= frac{1}{4} \ frac{x}{2}=arccos frac{1}{4} +n,neZ \ x= frac{ pi }{3} +2 pi n,neZ$
$3) cos frac{-3}{2}x=1 \ frac{-3}{2}x=2 pi n,neZ (* frac{-2}{3}) \ x= frac{-4}{3} pi n,neZ$