Упр.9 ГДЗ Колмогоров 10-11 класс (Алгебра)

а) Используем формулу косинуса разности:

$$\frac{\cos \frac{\pi}{15}\cos \frac{4\pi}{15}-\sin \frac{4\pi}{15}\sin \frac{\pi}{15}} {\cos 0{,}3\pi \sin 0{,}2\pi+\sin 0{,}3\pi \cos 0{,}2\pi} = \frac{\cos \left(\frac{\pi}{15}+\frac{4\pi}{15}\right)}{\sin \left(0{,}2\pi+0{,}3\pi\right)} = \frac{\cos \frac{\pi}{3}}{\sin \frac{\pi}{2}} = \frac{\frac12}{1} = \frac12.$$

б) Используем формулу тангенса разности:

$$\frac{\tg \frac{2\pi}{3}-\tg \frac{5\pi}{12}}{1+\tg \frac{2\pi}{3}\tg \frac{5\pi}{12}} = \tg \left(\frac{2\pi}{3}-\frac{5\pi}{12}\right) = \tg \frac{3\pi}{12} = \tg \frac{\pi}{4} = 1.$$

в) Используем формулу тангенса суммы:

$$\frac{\tg \frac{\pi}{10}+\tg \frac{3\pi}{20}}{1-\tg \frac{\pi}{10}\tg \frac{3\pi}{20}} = \tg \left(\frac{\pi}{10}+\frac{3\pi}{20}\right) = \tg \frac{5\pi}{20} = \tg \frac{\pi}{4} = 1.$$

г) Используем формулу синуса разности:

$$\frac{\sin \frac{5\pi}{18}\cos \frac{\pi}{9}-\sin \frac{\pi}{9}\cos \frac{5\pi}{18}} {\sin \frac{5\pi}{12}\sin \frac{7\pi}{12}-\cos \frac{5\pi}{12}\cos \frac{7\pi}{12}} = \frac{\sin \left(\frac{5\pi}{18}-\frac{\pi}{9}\right)} {-\cos \left(\frac{5\pi}{12}+\frac{7\pi}{12}\right)} = \frac{\sin \frac{\pi}{6}}{-\cos \pi} = \frac{\frac12}{1} = \frac12.$$

Ответ: а) $$\frac12$$; б) $$1$$; в) $$1$$; г) $$\frac12$$.