Упр.617 ГДЗ Алимов 10-11 класс (Алгебра)

1) arctg(сtg 5пи/6);
2) arctg(сtg 3пи/4);
3) arctg(2sin 5пи/6);
4) arctg(2sin пи/3).

1. $$\operatorname{arctg}\left(\operatorname{ctg}\frac{5\pi}{6}\right)=\operatorname{arctg}\left(\operatorname{ctg}\left(\frac{\pi}{2}+\frac{\pi}{3}\right)\right)=\operatorname{arctg}\left(-\operatorname{tg}\frac{\pi}{3}\right)=-\operatorname{arctg}\left(\operatorname{tg}\frac{\pi}{3}\right)=-\frac{\pi}{3}.$$
2. $$\operatorname{arctg}\left(\operatorname{ctg}\frac{3\pi}{4}\right)=\operatorname{arctg}\left(\operatorname{ctg}\left(\frac{\pi}{2}+\frac{\pi}{4}\right)\right)=\operatorname{arctg}\left(-\operatorname{tg}\frac{\pi}{4}\right)=-\operatorname{arctg}\left(\operatorname{tg}\frac{\pi}{4}\right)=-\frac{\pi}{4}.$$
3. $$\operatorname{arctg}\left(2\sin\frac{5\pi}{6}\right)=\operatorname{arctg}\left(2\sin\left(\pi-\frac{\pi}{6}\right)\right)=\operatorname{arctg}\left(2\sin\frac{\pi}{6}\right)=\operatorname{arctg}\left(2\cdot\frac12\right)=\operatorname{arctg}1=\frac{\pi}{4}.$$
4. $$\operatorname{arctg}\left(2\sin\frac{\pi}{3}\right)=\operatorname{arctg}\left(2\cdot\frac{\sqrt3}{2}\right)=\operatorname{arctg}\sqrt3=\frac{\pi}{3}.$$

Ответ

$$-\frac{\pi}{3};\ -\frac{\pi}{4};\ \frac{\pi}{4};\ \frac{\pi}{3}.$$