Упр.22.3 ГДЗ Мордкович 10-11 класс (Алгебра)

• а) $$\sin\frac{\pi}{5}-\sin\frac{\pi}{10}$$; б) $$\sin\frac{\pi}{3}+\sin\frac{\pi}{4}$$; в) $$\sin\frac{\pi}{6}+\sin\frac{\pi}{7}$$; г) $$\sin\frac{\pi}{3}-\sin\frac{\pi}{11}$$.

Воспользуемся формулами суммы и разности синусов:

$$\sin \alpha-\sin \beta=2\cos \frac{\alpha+\beta}{2}\sin \frac{\alpha-\beta}{2},$$

$$\sin \alpha+\sin \beta=2\sin \frac{\alpha+\beta}{2}\cos \frac{\alpha-\beta}{2}.$$

1. $$\sin \frac{\pi}{5}-\sin \frac{\pi}{10}=2\cos \frac{\frac{\pi}{5}+\frac{\pi}{10}}{2}\sin \frac{\frac{\pi}{5}-\frac{\pi}{10}}{2}=2\cos \frac{3\pi}{20}\sin \frac{\pi}{20}.$$

2. $$\sin \frac{\pi}{3}+\sin \frac{\pi}{4}=2\sin \frac{\frac{\pi}{3}+\frac{\pi}{4}}{2}\cos \frac{\frac{\pi}{3}-\frac{\pi}{4}}{2}=2\sin \frac{7\pi}{24}\cos \frac{\pi}{24}.$$

3. $$\sin \frac{\pi}{6}+\sin \frac{\pi}{7}=2\sin \frac{\frac{\pi}{6}+\frac{\pi}{7}}{2}\cos \frac{\frac{\pi}{6}-\frac{\pi}{7}}{2}=2\sin \frac{13\pi}{84}\cos \frac{\pi}{84}.$$

4. $$\sin \frac{\pi}{3}-\sin \frac{\pi}{11}=2\cos \frac{\frac{\pi}{3}+\frac{\pi}{11}}{2}\sin \frac{\frac{\pi}{3}-\frac{\pi}{11}}{2}=2\cos \frac{7\pi}{33}\sin \frac{4\pi}{33}.$$

Ответ: $$2\cos \frac{3\pi}{20}\sin \frac{\pi}{20};\ 2\sin \frac{7\pi}{24}\cos \frac{\pi}{24};\ 2\sin \frac{13\pi}{84}\cos \frac{\pi}{84};\ 2\cos \frac{7\pi}{33}\sin \frac{4\pi}{33}.$$