Упр.24.16 ГДЗ Мордковича 10 класс профильный уровень (Алгебра)

1. Вычислите:
   а) $$\sin\frac{\pi}{5}\cos\frac{\pi}{20}+\cos\frac{\pi}{5}\sin\frac{\pi}{20}$$
   б) $$\cos\frac{2\pi}{7}\cos\frac{5\pi}{7}-\sin\frac{2\pi}{7}\sin\frac{5\pi}{7}$$
   в) $$\sin\frac{\pi}{12}\cos\frac{11\pi}{12}+\cos\frac{\pi}{12}$$
   г) $$\cos\frac{2\pi}{15}\cos\frac{\pi}{5}-\sin\frac{2\pi}{15}\sin\frac{\pi}{5}$$

Используем формулы сложения:

$$\sin \alpha \cos \beta + \cos \alpha \sin \beta = \sin(\alpha+\beta),$$

$$\cos \alpha \cos \beta — \sin \alpha \sin \beta = \cos(\alpha+\beta).$$

1. $$\sin \frac{\pi}{5}\cos \frac{\pi}{20}+\cos \frac{\pi}{5}\sin \frac{\pi}{20}=\sin\left(\frac{\pi}{5}+\frac{\pi}{20}\right)=\sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}.$$

2. $$\cos \frac{2\pi}{7}\cos \frac{5\pi}{7}-\sin \frac{2\pi}{7}\sin \frac{5\pi}{7}=\cos\left(\frac{2\pi}{7}+\frac{5\pi}{7}\right)=\cos \pi=-1.$$

3. $$\sin \frac{\pi}{12}\cos \frac{11\pi}{12}+\cos \frac{\pi}{12}\sin \frac{11\pi}{12}=\sin\left(\frac{\pi}{12}+\frac{11\pi}{12}\right)=\sin \pi=0.$$

4. $$\cos \frac{2\pi}{15}\cos \frac{\pi}{5}-\sin \frac{2\pi}{15}\sin \frac{\pi}{5}=\cos\left(\frac{2\pi}{15}+\frac{\pi}{5}\right)=\cos \frac{\pi}{3}=\frac{1}{2}.$$

Ответ: а) $$\frac{\sqrt{2}}{2}$$; б) $$-1$$; в) $$0$$; г) $$\frac{1}{2}$$.