Упр.640 ГДЗ Никольский Потапов 9 класс (Алгебра)

1. а) $$\cos\frac{\pi}{5}-\cos\frac{\pi}{4}$$; б) $$\sin\frac{\pi}{14}+\sin\frac{\pi}{3}$$; в) $$\sin\frac{\pi}{3}-\sin\frac{\pi}{4}$$; г) $$\cos\frac{\pi}{10}+\cos\frac{\pi}{5}$$

Воспользуемся формулами суммы и разности синусов и косинусов:

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

а)

$$\cos \frac{\pi}{5}-\cos \frac{\pi}{4} =-2\sin \frac{\frac{\pi}{5}+\frac{\pi}{4}}{2}\cdot \sin \frac{\frac{\pi}{5}-\frac{\pi}{4}}{2}$$
$$=-2\sin \frac{9\pi}{40}\cdot \sin \left(-\frac{\pi}{40}\right) =2\sin \frac{9\pi}{40}\cdot \sin \frac{\pi}{40}$$

б)

$$\sin \frac{\pi}{14}+\sin \frac{\pi}{3} =2\sin \frac{\frac{\pi}{14}+\frac{\pi}{3}}{2}\cdot \cos \frac{\frac{\pi}{14}-\frac{\pi}{3}}{2}$$
$$=2\sin \frac{17\pi}{84}\cdot \cos \left(-\frac{11\pi}{84}\right) =2\sin \frac{17\pi}{84}\cdot \cos \frac{11\pi}{84}$$

в)

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

г)

$$\cos \frac{\pi}{10}+\cos \frac{\pi}{5} =2\cos \frac{\frac{\pi}{10}+\frac{\pi}{5}}{2}\cdot \cos \frac{\frac{\pi}{10}-\frac{\pi}{5}}{2}$$
$$=2\cos \frac{3\pi}{20}\cdot \cos \left(-\frac{3\pi}{20}\right) =2\cos \frac{3\pi}{20}\cdot \cos \frac{3\pi}{20}$$

Ответ

а) $$2\sin \frac{9\pi}{40}\cdot \sin \frac{\pi}{40}$$; б) $$2\sin \frac{17\pi}{84}\cdot \cos \frac{11\pi}{84}$$; в) $$2\sin \frac{\pi}{24}\cdot \cos \frac{7\pi}{24}$$; г) $$2\cos \frac{3\pi}{20}\cdot \cos \frac{3\pi}{20}$$.