Question

Помогите упростить .............

Answer 1

Ответ:

$-\frac{1}{8}$

Пошаговое объяснение:

$cos(\frac{2\pi}{7})cos(\frac{3\pi}{7})cos(\frac{6\pi}{7}) =-cos(\frac{2\pi}{7})cos(\frac{3\pi}{7})cos(\frac{\pi}{7}) = -\frac{1}{2} * \frac{sin(\frac{2\pi}{7})cos(\frac{2\pi}{7})cos(\frac{3\pi}{7})}{sin{(\frac{\pi}{7}})}=$$= -\frac{1}{4} * \frac{sin(\frac{4\pi}{7})cos(\frac{3\pi}{7})}{sin{(\frac{\pi}{7}})} = -\frac{1}{4} * \frac{sin( \pi -\frac{3\pi}{7})cos(\frac{3\pi}{7})}{sin{(\frac{\pi}{7}})} =-\frac{1}{4} * \frac{sin( \frac{3\pi}{7})cos(\frac{3\pi}{7})}{sin{(\frac{\pi}{7}})} =$

$= -\frac{1}{8} * \frac{sin( \frac{6\pi}{7})}{sin{(\frac{\pi}{7}})} = -\frac{1}{8}$

Answer 2

Домножим числитель и знаменатель на 8sin(2π/7)*sin(3π/7)*sin(6π/7); воспользуемся формулой 2sin(α)*сosα=sin(α) и формулами  приведения, получим

(2sin(2π/7)*сos(2π/7))*(2sin(3π/7)*сos2(3π/7))*

(2sin(6π/7)сos2(6π/7))/( 8sin(2π/7)*sin(3π/7)*sin(6π/7)=

sin(4π/7)*sin(6π/7)*sin(12π/7)/( 8sin(2π/7)*sin(3π/7)*sin(6π/7)=

sin(4π/7)*sin(12π/7)/( 8sin(2π/7)*sin(3π/7))=

sin(π-3π/7)*sin(2π-2π/7)/( 8sin(2π/7)*sin(3π/7))=

sin(3π/7)*sin(-2π/7)/( 8sin(2π/7)*sin(3π/7))=-1/8