Question

1) sinx+sin2x+sin3x+sin4x=0 2) cosx+cos2x+cos3x+cos4x=0

Answer 1

$sinx+sin2x+sin3x+sin4x=0 \\\\ 2sin \frac{x+4x}{2}cos \frac{x-4x}{2} +2sin \frac{2x+3x}{2}cos \frac{2x-3x}{2}=0 \\\\ sin \frac{5x}{2}cos \frac{3x}{2} +sin \frac{5x}{2}cos \frac{x}{2}=0 \\\\ sin \frac{5x}{2}(cos \frac{3x}{2} +cos \frac{x}{2})=0 \\\\ sin \frac{5x}{2}=0 \\ \frac{5x}{2}= \pi n \\ x= \frac{2}{5} \pi n,n\in Z \\\\$

$cos \frac{3x}{2} +cos \frac{x}{2}=0 \\ 2cos \frac{\frac{3x}{2}+ \frac{x}{2}}{2} cos \frac{\frac{3x}{2}- \frac{x}{2}}{2} =0 \\ cosxcos \frac{x}{2} =0 \\ cosx=0 \\ x= \frac{ \pi }{2} + \pi n,n\in Z \\ cos \frac{x}{2} =0 \\ \frac{x}{2} = \frac{ \pi }{2} + \pi n,n\in Z \\ x= \pi +2 \pi n,\n\in Z$

$cosx+cos2x+cos3x+cos4x=0 \\\\ 2cos \frac{x+4x}{2} cos \frac{x-4x}{2} +2cos \frac{2x+3x}{2} cos \frac{2x-3x}{2} =0 \\\\ cos \frac{5x}{2} cos \frac{3x}{2} +cos \frac{5x}{2} cos \frac{x}{2} =0 \\\\ cos \frac{5x}{2} (cos \frac{3x}{2} + cos \frac{x}{2}) =0 \\\\ cos \frac{5x}{2}=0 \\ \frac{5x}{2}= \frac{ \pi }{2} + \pi n \\ x= \frac{ \pi }{5} + \frac{2}{5} \pi n, n\in Z$

$cos \frac{3x}{2} + cos \frac{x}{2}=0$ решали в уравнении