Question

16 cos pi/ 18 ∙ cos pi/ 6 ∙ cos 5pi/18 ∙ cos 7pi/ 18
помогите пожалуйста с пояснением​

Answer 1

Ответ:

3

Пошаговое объяснение:

Запишем

$\displaystyle\frac{\pi }{6} = \displaystyle\frac{{3\pi }}{{18}}.$

Используя формулу преобразования произведения косинусов

$\cos \alpha \cos \beta = \displaystyle\frac{1}{2}\left( {\cos (\alpha - \beta ) + \cos (\alpha + \beta )} \right),$

получаем:

$2\cos \displaystyle\frac{{3\pi }}{{18}}\cos \displaystyle\frac{\pi }{{18}} = \cos \left( {\displaystyle\frac{{3\pi }}{{18}} - \displaystyle\frac{\pi }{{18}}} \right) + \cos \left( {\displaystyle\frac{{3\pi }}{{18}} + \displaystyle\frac{\pi }{{18}}} \right) = \cos \displaystyle\frac{\pi }{9} + \cos \displaystyle\frac{{2\pi }}{9};$

$2\cos \displaystyle\frac{{7\pi }}{{18}}\cos \displaystyle\frac{{5\pi }}{{18}} = \cos \left( {\displaystyle\frac{{7\pi }}{{18}} - \displaystyle\frac{{5\pi }}{{18}}} \right) + \cos \left( {\displaystyle\frac{{7\pi }}{{18}} + \displaystyle\frac{{5\pi }}{{18}}} \right) = \cos \displaystyle\frac{\pi }{9} + \cos \displaystyle\frac{{2\pi }}{3}.$

Отсюда

$16\cos \displaystyle\frac{\pi }{{18}}\cos \displaystyle\frac{\pi }{6}\cos \displaystyle\frac{{5\pi }}{{18}}\cos \displaystyle\frac{{7\pi }}{{18}} = 4\left( {\cos \displaystyle\frac{\pi }{9} + \cos \displaystyle\frac{{2\pi }}{9}} \right)\left( {\cos \displaystyle\frac{\pi }{9} - \displaystyle\frac{1}{2}} \right) = \\\\=4{\cos ^2}\displaystyle\frac{\pi }{9} + 4\cos \displaystyle\frac{{2\pi }}{9}\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{{2\pi }}{9}.$

По формуле понижения степени

${\cos ^2}\alpha = \displaystyle\frac{{\cos 2\alpha + 1}}{2}$

получаем:

$4{\cos ^2}\displaystyle\frac{\pi }{9} + 4\cos \displaystyle\frac{{2\pi }}{9}\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{{2\pi }}{9} = \\\\=2\cos \displaystyle\frac{{2\pi }}{9} + 2 + 4\cos \displaystyle\frac{{2\pi }}{9}\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{{2\pi }}{9} =\\\\ =2 + 4\cos \displaystyle\frac{{2\pi }}{9}\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{\pi }{9}.$

Снова воспользуемся формулой произведения косинусов

$2 + 4\cos \displaystyle\frac{{2\pi }}{9}\cos \displaystyle\frac{\pi }{9} - 2\cos \displaystyle\frac{\pi }{9} = 2 + 2\left( {\cos \left( {\displaystyle\frac{{2\pi }}{9} - \displaystyle\frac{\pi }{9}} \right) + \cos \left( {\displaystyle\frac{{2\pi }}{9} + \displaystyle\frac{\pi }{9}} \right)} \right) - 2\cos \displaystyle\frac{\pi }{9} =\\ =2 + 2\cos \displaystyle\frac{\pi }{9} + 2\cos \displaystyle\frac{\pi }{3} - 2\cos \displaystyle\frac{\pi }{9} = 2 + 2 \cdot \displaystyle\frac{1}{2} = 3.$