Question

упростить sinx+sin2x+ sin3x+ sin4x

Answer 1

$(sinx+sin3x)+(sin2x+sin4x)=0$
$2sinx frac{x+3x}{2} * cos frac{x-3x}{2} + 2sin frac{2x+4x}{2} * cos frac{2x-4x}{2} =0$
$2sin2x*cosx+2sin3x*cosx=0$
$2cosx(sin2x+sin3x)=0$
$cosx=0$
$x= frac{pi}{2} + pik$

$sin2x+sin3x=0$
$2sin frac{2x+3x}{2} * cos frac{2x-3x}{2}=0$
$2sin frac{5x}{2}=0; cos frac{x}{2}=0$
$frac{5x}{2}=pik; frac{x}{2}= frac{pi}{2}+pik$
$5x=2pik; x=pi+2pik$
$x= frac{2pik}{5}$