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

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. $$\operatorname{arctg}\left(2\sin\frac{\pi}{3}\right)$$.

Используем определения и значения тригонометрических функций.

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