Упр.22.4 ГДЗ Мордкович 10-11 класс (Алгебра)

a) cos РїРё/10 — cos РїРё/20;
6) cos 11РїРё/12 + cos 3РїРё/4;
РІ) cos РїРё/5 — cos РїРё/11;
Рі) cos 3РїРё/8 + cos 5РїРё/4.

1. $$\cos \frac{\pi}{10}-\cos \frac{\pi}{20}=-2\sin \frac{\frac{\pi}{10}+\frac{\pi}{20}}{2}\sin \frac{\frac{\pi}{10}-\frac{\pi}{20}}{2}$$
   $$=-2\sin \frac{3\pi}{40}\sin \frac{\pi}{40}.$$
2. $$\cos \frac{11\pi}{12}+\cos \frac{3\pi}{4}=2\cos \frac{\frac{11\pi}{12}+\frac{3\pi}{4}}{2}\cos \frac{\frac{11\pi}{12}-\frac{3\pi}{4}}{2}$$
   $$=2\cos \frac{5\pi}{6}\cos \frac{\pi}{12}=-2\cos \frac{\pi}{6}\cos \frac{\pi}{12}$$
   $$=-\sqrt{3}\cos \frac{\pi}{12}.$$
3. $$\cos \frac{\pi}{5}-\cos \frac{\pi}{11}=-2\sin \frac{\frac{\pi}{5}+\frac{\pi}{11}}{2}\sin \frac{\frac{\pi}{5}-\frac{\pi}{11}}{2}$$
   $$=-2\sin \frac{8\pi}{55}\sin \frac{3\pi}{55}.$$
4. $$\cos \frac{3\pi}{8}+\cos \frac{5\pi}{4}=2\cos \frac{\frac{3\pi}{8}+\frac{5\pi}{4}}{2}\cos \frac{\frac{3\pi}{8}-\frac{5\pi}{4}}{2}$$
   $$=2\cos \frac{13\pi}{16}\cos \left(-\frac{7\pi}{16}\right)=2\cos \frac{13\pi}{16}\cos \frac{7\pi}{16}.$$

Ответ

а) $$-2\sin \frac{3\pi}{40}\sin \frac{\pi}{40}$$;
б) $$-\sqrt{3}\cos \frac{\pi}{12}$$;
в) $$-2\sin \frac{8\pi}{55}\sin \frac{3\pi}{55}$$;
г) $$2\cos \frac{13\pi}{16}\cos \frac{7\pi}{16}$$.