Question

sin5x+sin2x+sin3x+sin4x=0

Answer 1

$sin5x+sin2x+sin3x+sin4x=0$
$(sin5x+sin3x)+(sin2x+sin4x)=0$
$2sinfrac{5x+3x}{2} cosfrac{5x-3x}{2} +2sinfrac{2x+4x}{2} cosfrac{2x-4x}{2} =0$
$sin4x cos x+sin3x cos (-x)=0$
$sin4x cos x+sin3x cos x=0$
$cos x(sin4x +sin3x )=0$
$left[begin{array}{l} cos x=0\ sin4x +sin3x=0 end{array}$
$left[begin{array}{l} cos x=0\ 2sin frac{4x+3x}{2} cos frac{4x-3x}{2} =0 end{array}$
$left[begin{array}{l} cos x=0 \ sin frac{7x}{2}=0 \ cos frac{x}{2} =0 end{array}$
$left[begin{array}{l} x= frac{ pi }{2}+pi n \\ frac{7x}{2}= pi n \\ frac{x}{2} = frac{ pi }{2}+ pi n end{array} Rightarrow left[begin{array}{l} x_1= frac{ pi }{2}+pi n \\ x_2= frac{ 2pi n}{7} \\ x_3= pi + 2pi n end{array}, nin Z$