Страница 184 номер 617, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Вычислить: 1) \(\operatorname{arctg}\left(\operatorname{ctg}\frac{5\pi}{6}\right)\); 2) \(\operatorname{arctg}\left(\operatorname{ctg}\frac{3\pi}{4}\right)\); 3) \(\operatorname{arctg}\left(2 \sin\frac{5\pi}{6}\right)\); 4) \(\operatorname{arctg}\left(2 \sin\frac{\pi}{3}\right)\).
1) \(\operatorname{arctg} \left( \operatorname{ctg} \frac{5\pi}{6} \right) = \operatorname{arctg} \left( \operatorname{ctg} \left( \frac{\pi}{2} + \frac{\pi}{3} \right) \right) = \operatorname{arctg} \left(-\operatorname{tg} \frac{\pi}{3} \right) = - \frac{\pi}{3}\);
2) \(\operatorname{arctg} \left( \operatorname{ctg} \frac{3\pi}{4} \right) = \operatorname{arctg} \left( \operatorname{ctg} \left( \frac{\pi}{2} + \frac{\pi}{4} \right) \right) = \operatorname{arctg} \left(-\operatorname{tg} \frac{\pi}{4} \right) = - \frac{\pi}{4}\);
3) \(\operatorname{arctg} \left( 2\sin\frac{5\pi}{6} \right) = \operatorname{arctg} \left( 2 \cdot \sin\left( \pi - \frac{\pi}{6} \right) \right) = \operatorname{arctg} \left( 2 \cdot \frac{1}{2} \right) = \frac{\pi}{4}\);
4) \(\operatorname{arctg} \left( 2\sin\frac{\pi}{3} \right) = \operatorname{arctg} \left( 2 \cdot \frac{\sqrt{3}}{2} \right) = \frac{\pi}{3}\).
