sin(x)-sin(2x)+sin(5x)+sin(8x)=0

Question

sin(x)-sin(2x)+sin(5x)+sin(8x)=0

Answer 1

(sinx+sin5x)+(sin8x-sin2x)=0
2sin3xcos2x+2sin3xcos5x=0
2sin3x*(cos2x+cos5x)=0
4sin3x*cos7x/2cos3x/2=0
sin3x=0⇒3x=πn⇒x=πn/3
cos7x/2=0⇒7x/2=π/2+2πk⇒x=π/7+4k/7,k∈z
cos3x/2=0⇒3x/2=π/2+2πk⇒x=π+4m/3,m∈z