Question

решите уравнение sinx=sin6x

Answer 1

$sin x=sin6x;\ sin x=2sin3xcos3x;\ sin3x=sin(2x+x)=sin2xcos x+sin xcos2x=\ =2sin xcos^2x+sin xcos 2x=sin x(2cos^2x+cos2x);\ sin x=sin x(2cos^2x+cos2x)cos3x;\ cos3x=cos(2x+x)=cos2xcos x-sin2xsin x=\ =cos xcos 2x-2sin xcos xsin x=cos x(cos2x+1-2sin x-1)=\ =cos x(2cos2x-1)\ 1=(2cos^2x-1+cos2x+1)cos x(2cos2x-1);\ 1=cos x(2cos2x+1)(2cos2x-1);\ 1=cos x(4cos^22x-1);\ 1=cos x(4(2cos^2x-1)^2-1);\ 1=cos x(16cos^4x-16cos^2x+4-1);\ cos x=t;$
$1=t(16t^4-16t^2+3);\ 16t^5-16t^3+3t-1=0;$