Question

sin(49*pi/12)*(sin^4(59*pi/24) - sin^4(49*pi/24)).
Как найти значение этого выражения?

Answer 1

$=sin(4 pi + frac{pi}{12})*(sin^4(2pi+frac{11 pi}{24}) -sin^4(2pi +frac{pi}{24})) = \ sin(frac{pi}{12})*(sin^4(frac{11 pi}{24}) -sin^4(frac{pi}{24})) = \ sin(frac{pi}{12})*(sin^4( frac{pi}{2} - frac{pi}{24}) -sin^4(frac{pi}{24})) = \ sin(frac{pi}{12})*(cos^4(frac{pi}{24}) -sin^4(frac{pi}{24})) = \ sin(frac{pi}{12})*(cos^2(frac{pi}{24}) -sin^2(frac{pi}{24}))*(cos^2(frac{pi}{24}) +sin^2(frac{pi}{24})) = \ sin(frac{pi}{12}) *cos(2*frac{pi}{24})*1=$
$=sin(frac{pi}{12}) *cos(frac{pi}{12}) = frac{2*sin(frac{pi}{12}) *cos(frac{pi}{12})}{2} = frac{sin(frac{pi}{6})}{2} = frac{frac{1}{2} }{2} =frac{1}{4} =0,25$