Question

Найдите значение выражения
$32(sin^{3} a-cos^{3} a)$
если
$sina-cosa=0.25$

Answer 1

Формула разность кубов:

sin^3(a) - cos^3(a) = (sin(a) - cos(a))*(sin^2(a) + sin(a)*cos(a) + cos^2(a)) = (1/2)*(1 + sin(a)*cos(a)),

так как sin(a) - cos(a) = 1/2 и sin^2(a) + cos^2(a) = 1.

Вычислим sin(a)*cos(a).

(1/2)^2 = (sin(a) - cos(a))^2 = sin^2(a) - 2*sin(a)*cos(a) + cos^2(a) = 1 - 2*sin(a)*cos(a)

2*sin(a)*cos(a) = 3/4

sin(a)*cos(a) = 3/8

Получаем:

sin^3(a) - cos^3(a) = (1/2)*(1 + sin(a)*cos(a)) = (1/2)*(1 + 3/8) = 11/16