Question

ctg х = 1/3
Найти: sin 4x

Answer 1

$ctgx=frac{1}{3}\ frac{cosx}{sinx}=frac{1}{3}\ 3cosx=sinx\ 3sqrt{1-sin^2x}=sinx\ 9(1-sin^2x)=sin^2x\ 10sin^2x=9\ sinx=frac{3}{sqrt{10}}\ x=arcsinfrac{3}{sqrt{10}}\ x=arcos(frac{1}{sqrt{10}})\ sin4x=2sin2xcos2x=4sinxcosx(2cos^2x-1)=\ 4sin(arcsinfrac{3}{sqrt{10}})*cos(arcos(frac{1}{sqrt{10}}))(2cos(arcos(frac{1}{sqrt{10}}))^2-1)=frac{12}{sqrt{10}}*frac{1}{sqrt{10}}(2*frac{1}{10}-1)=frac{12}{10}(frac{2}{10}-1)=-frac{24}{25}$