Question

Вычислите: sin5p/24*sinp/24-cosp/24*cos5p/24

Answer 1

$Sin\frac{5\pi }{24}*Sin\frac{\pi }{24}-Cos\frac{\pi }{24}*Cos\frac{5\pi }{24}=-Cos(\frac{5\pi }{24}+\frac{\pi }{24})=-Cos\frac{6\pi }{24}=-Cos\frac{\pi }{4}=-\frac{\sqrt{2}}{2}$

Answer 2

Давайте разберёмся с этим выражением:

sin(5π/24) * sin(π/24) - cos(π/24) * cos(5π/24)

Мы можем воспользоваться тригонометрическими формулами для произведения синусов и косинусов углов:

1. Произведение синусов: sin(a) * sin(b) = (1/2) * [cos(a - b) - cos(a + b)]

2. Произведение косинусов: cos(a) * cos(b) = (1/2) * [cos(a - b) + cos(a + b)]

Применяя эти формулы, получаем:

(sin(5π/24) * sin(π/24)) - (cos(π/24) * cos(5π/24)) = (1/2) * [cos(5π/24 - π/24) - cos(5π/24 + π/24)] - (1/2) * [cos(π/24 - 5π/24) + cos(π/24 + 5π/24)] = (1/2) * [cos(4π/24) - cos(6π/24)] - (1/2) * [-cos(4π/24) + cos(6π/24)] = (1/2) * [cos(π/6) - cos(π/4)] - (1/2) * [-cos(π/6) + cos(π/4)] = (1/2) * [√3/2 - √2/2] - (1/2) * [-√3/2 + √2/2] = (1/2) * (√3/2 - √2/2 + √3/2 - √2/2) = √3/2 - √2/2

Итак, значение выражения равно: √3/2 - √2/2.