Question

sin5x+sinx=5sin3x решите уравнение

Answer 1

$displaystyle sin5x+sin x=5sin3xquad quad quad quad bigg(sin a+sin b=2sinfrac{a+b}2cosfrac{a-b}2bigg)\\2sinfrac{5x+x}2cosfrac{5x-x}2=5sin3x\\2sin3xcos2x=5sin3x\\2sin3xcos2x-5sin3x=0\\sin3x(2cos2x-5)=0\\\ left[begin{array}{ccc}sin3x=0\2cos2x-5=0end{array}right = textgreater left[begin{array}{ccc}3x=pi n; ,,nin Z\cos2x=2,5end{array}right= textgreater left[begin{array}{ccc}displaystyle x=frac{pi n}3;,,in Z\varnothingend{array}right$

Ответ:  $boxed{displaystyle x=frac{pi n}3;,,in Z}$