Упр.26.15 ГДЗ Мордковича 10 класс профильный уровень (Алгебра)

1. а) $$\frac{2\cos\frac{11\pi}{5}+8\sin\frac{13\pi}{10}}{\cos\frac{\pi}{5}}$$; б) $$\frac{5\sin\frac{5\pi}{7}+2\cos\frac{25\pi}{14}}{\sin\frac{2\pi}{7}}$$

a) $$\frac{2\cos \frac{11\pi}{5}+8\sin \frac{13\pi}{10}}{\cos \frac{\pi}{5}}$$

Преобразуем аргументы тригонометрических функций:

$$\cos \frac{11\pi}{5}=\cos \left(2\pi+\frac{\pi}{5}\right)=\cos \frac{\pi}{5},$$

$$\sin \frac{13\pi}{10}=\sin \left(\frac{3\pi}{2}-\frac{\pi}{5}\right)=-\cos \frac{\pi}{5}.$$

Тогда

$$\frac{2\cos \frac{\pi}{5}+8\left(-\cos \frac{\pi}{5}\right)}{\cos \frac{\pi}{5}}=\frac{-6\cos \frac{\pi}{5}}{\cos \frac{\pi}{5}}=-6.$$

б) $$\frac{5\sin \frac{5\pi}{7}+2\cos \frac{25\pi}{14}}{\sin \frac{2\pi}{7}}$$

Преобразуем:

$$\sin \frac{5\pi}{7}=\sin \left(\pi-\frac{2\pi}{7}\right)=\sin \frac{2\pi}{7},$$

$$\cos \frac{25\pi}{14}=\cos \left(\frac{3\pi}{2}+\frac{2\pi}{7}\right)=\sin \frac{2\pi}{7}.$$

Тогда

$$\frac{5\sin \frac{2\pi}{7}+2\sin \frac{2\pi}{7}}{\sin \frac{2\pi}{7}}=\frac{7\sin \frac{2\pi}{7}}{\sin \frac{2\pi}{7}}=7.$$

Ответ: а) $$-6$$; б) $$7$$.