Упр.27.25 ГДЗ Мордковича 10 класс профильный уровень (Алгебра)

Рассмотрим вариант решения задания из учебника Мордкович, Семенов 10 класс, Мнемозина: a) cos пи/33 cos 2пи/33 cos 4пи/33 cos 8пи/33 cos 16пи/33 ; б) cos пи/65 cos 2пи/65 cos 4пи/65 cos 8пи/65 cos 16пи/65 cos 32пи/65 .

Используем формулу

$$2\sin x\cos x=\sin 2x.$$

a)

$$ \cos \frac{\pi}{33}\cdot \cos \frac{2\pi}{33}\cdot \cos \frac{4\pi}{33}\cdot \cos \frac{8\pi}{33}\cdot \cos \frac{16\pi}{33} $$
$$ =\frac{1}{2\sin \frac{\pi}{33}}\cdot \left(2\sin \frac{\pi}{33}\cos \frac{\pi}{33}\right)\cos \frac{2\pi}{33}\cos \frac{4\pi}{33}\cos \frac{8\pi}{33}\cos \frac{16\pi}{33} $$
$$ =\frac{1}{2\sin \frac{\pi}{33}}\cdot \sin \frac{2\pi}{33}\cos \frac{2\pi}{33}\cos \frac{4\pi}{33}\cos \frac{8\pi}{33}\cos \frac{16\pi}{33} $$
$$ =\frac{1}{4\sin \frac{\pi}{33}}\cdot \sin \frac{4\pi}{33}\cos \frac{4\pi}{33}\cos \frac{8\pi}{33}\cos \frac{16\pi}{33} $$
$$ =\frac{1}{8\sin \frac{\pi}{33}}\cdot \sin \frac{8\pi}{33}\cos \frac{8\pi}{33}\cos \frac{16\pi}{33} $$
$$ =\frac{1}{16\sin \frac{\pi}{33}}\cdot \sin \frac{16\pi}{33}\cos \frac{16\pi}{33} =\frac{1}{32\sin \frac{\pi}{33}}\sin \frac{32\pi}{33}. $$
$$ \sin \frac{32\pi}{33}=\sin\left(\pi-\frac{\pi}{33}\right)=\sin \frac{\pi}{33}, $$
значит
$$ \cos \frac{\pi}{33}\cdot \cos \frac{2\pi}{33}\cdot \cos \frac{4\pi}{33}\cdot \cos \frac{8\pi}{33}\cdot \cos \frac{16\pi}{33} =\frac{1}{32}. $$

б)

$$ \cos \frac{\pi}{65}\cdot \cos \frac{2\pi}{65}\cdot \cos \frac{4\pi}{65}\cdot \cos \frac{8\pi}{65}\cdot \cos \frac{16\pi}{65}\cdot \cos \frac{32\pi}{65} $$
$$ =\frac{1}{2\sin \frac{\pi}{65}}\cdot \left(2\sin \frac{\pi}{65}\cos \frac{\pi}{65}\right)\cos \frac{2\pi}{65}\cos \frac{4\pi}{65}\cos \frac{8\pi}{65}\cos \frac{16\pi}{65}\cos \frac{32\pi}{65} $$
$$ =\frac{1}{2\sin \frac{\pi}{65}}\cdot \sin \frac{2\pi}{65}\cos \frac{2\pi}{65}\cos \frac{4\pi}{65}\cos \frac{8\pi}{65}\cos \frac{16\pi}{65}\cos \frac{32\pi}{65} $$
$$ =\frac{1}{4\sin \frac{\pi}{65}}\cdot \sin \frac{4\pi}{65}\cos \frac{4\pi}{65}\cos \frac{8\pi}{65}\cos \frac{16\pi}{65}\cos \frac{32\pi}{65} $$
$$ =\frac{1}{8\sin \frac{\pi}{65}}\cdot \sin \frac{8\pi}{65}\cos \frac{8\pi}{65}\cos \frac{16\pi}{65}\cos \frac{32\pi}{65} $$
$$ =\frac{1}{16\sin \frac{\pi}{65}}\cdot \sin \frac{16\pi}{65}\cos \frac{16\pi}{65}\cos \frac{32\pi}{65} $$
$$ =\frac{1}{32\sin \frac{\pi}{65}}\cdot \sin \frac{32\pi}{65}\cos \frac{32\pi}{65} =\frac{1}{64\sin \frac{\pi}{65}}\sin \frac{64\pi}{65}. $$
$$ \sin \frac{64\pi}{65}=\sin\left(\pi-\frac{\pi}{65}\right)=\sin \frac{\pi}{65}, $$
поэтому
$$ \cos \frac{\pi}{65}\cdot \cos \frac{2\pi}{65}\cdot \cos \frac{4\pi}{65}\cdot \cos \frac{8\pi}{65}\cdot \cos \frac{16\pi}{65}\cdot \cos \frac{32\pi}{65} =\frac{1}{64}. $$

Ответ

а) $$\frac{1}{32}$$; б) $$\frac{1}{64}$$.