Вычислите значение выр-ия: cos п/9 * сos 2п/9 - вопрос №12614109294989169 от kris411 04.04.2023 08:43

Question

Алгебра: Вычислите значение выр-ия: cos п/9 * сos 2п/9 * cos 4п/9 * cos 8п/9 * cos 16п/9 * cos 32п/9

kris411 · Answer

cosx cos2x cos4x cos8x cos16x cos32x =

= (2sinxcosx / 2sinx) cos2x cos4x cos8x cos16x cos32x =

= (sin2x / 2sinx) cos2x cos4x cos8x cos16x cos32x =

= (sin2x / 2sinx) (2sin2xcos2x / 2sin2x) cos4x cos8x cos16x cos32x =

= (sin4x / 4sinx) (2sin4xcos4x / 2sin4x) cos8x cos16x cos32x =

= (sin8x / 8sinx) (2sin8xcos8x / 2sin8x) cos16x cos32x =

= (sin16x / 16sinx) (2sin16xcos16x / 2sin16x) cos32x =

= (sin32x / 32sinx) (2sin32xcos32x / 2sin32x) =

= sin64x / 64sinx

x = π/9

$\frac{sin\frac{64\pi}{9}}{64sin\frac{\pi}{9}}=\frac{sin(7\pi+\frac{\pi}{9})}{64sin\frac{\pi}{9}}= \frac{sin7\pi cos\frac{\pi}{9}+cos7\pi sin\frac{\pi}{9}}{64sin\frac{\pi}{9}} = \frac{-sin\frac{\pi}{9}}{64sin\frac{\pi}{9}} = -\frac{1}{64}$