Упр.683 ГДЗ Никольский Потапов 9 класс (Алгебра)

а) sin 9пи/28•cos 5пи/28-sin 6пи/35•cos пи/35
б) cos 3пи/16•cos пи/16-cos 5пи/16•cos 3пи/16=v2/4

Воспользуемся формулами произведения в сумму:

$$\sin \alpha \cos \beta=\frac12\bigl(\sin(\alpha+\beta)+\sin(\alpha-\beta)\bigr),$$
$$\cos \alpha \cos \beta=\frac12\bigl(\cos(\alpha+\beta)+\cos(\alpha-\beta)\bigr).$$

а)

$$ \sin \frac{9\pi}{28}\cos \frac{5\pi}{28}-\sin \frac{6\pi}{35}\cos \frac{\pi}{35} $$
$$ =\frac12\left(\sin \frac{14\pi}{28}+\sin \frac{4\pi}{28}\right)-\frac12\left(\sin \frac{7\pi}{35}+\sin \frac{5\pi}{35}\right) $$
$$ =\frac12\left(\sin \frac{\pi}{2}+\sin \frac{\pi}{7}\right)-\frac12\left(\sin \frac{\pi}{5}+\sin \frac{\pi}{7}\right) $$
$$ =\frac12+\frac12\sin \frac{\pi}{7}-\frac12\sin \frac{\pi}{5}-\frac12\sin \frac{\pi}{7} =\frac12-\frac12\sin \frac{\pi}{5}. $$

б)

$$ \cos \frac{3\pi}{16}\cos \frac{\pi}{16}-\cos \frac{5\pi}{16}\cos \frac{3\pi}{16} $$
$$ =\frac12\left(\cos \frac{4\pi}{16}+\cos \frac{2\pi}{16}\right)-\frac12\left(\cos \frac{8\pi}{16}+\cos \frac{2\pi}{16}\right) $$
$$ =\frac12\left(\cos \frac{\pi}{4}+\cos \frac{\pi}{8}\right)-\frac12\left(\cos \frac{\pi}{2}+\cos \frac{\pi}{8}\right) $$
$$ =\frac12\cdot \frac{\sqrt2}{2}+\frac12\cos \frac{\pi}{8}-\frac12\cdot 0-\frac12\cos \frac{\pi}{8} =\frac{\sqrt2}{4}. $$

Ответ

а) $$\frac12-\frac12\sin \frac{\pi}{5}$$;
б) $$\frac{\sqrt2}{4}$$.