Question

Вычислите значение выр-ия:

cos п/9 * сos 2п/9 * cos 4п/9 * cos 8п/9 * cos 16п/9 * cos 32п/9

Answer 1

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

 

$frac{sinfrac{64pi}{9}}{64sinfrac{pi}{9}}=frac{sin(7pi+frac{pi}{9})}{64sinfrac{pi}{9}}= frac{sin7pi cosfrac{pi}{9}+cos7pi sinfrac{pi}{9}}{64sinfrac{pi}{9}} = frac{-sinfrac{pi}{9}}{64sinfrac{pi}{9}} = -frac{1}{64}$