17) ctg(7π/3); 18) tg(-4π/3); 19) tg(-7π/6); 20) tg(-8π/3)

Решение:

1. \( \operatorname{ctg}(\frac{7\pi}{3}) = \operatorname{ctg}(2\pi + \frac{\pi}{3}) = \operatorname{ctg}(\frac{\pi}{3}) = \frac{\sqrt{3}}{3} \)
2. \( \operatorname{tg}(-\frac{4\pi}{3}) = -\operatorname{tg}(\frac{4\pi}{3}) = -\operatorname{tg}(\pi + \frac{\pi}{3}) = -\operatorname{tg}(\frac{\pi}{3}) = -\sqrt{3} \)
3. \( \operatorname{tg}(-\frac{7\pi}{6}) = -\operatorname{tg}(\frac{7\pi}{6}) = -\operatorname{tg}(\pi + \frac{\pi}{6}) = -\operatorname{tg}(\frac{\pi}{6}) = -\frac{\sqrt{3}}{3} \)
4. \( \operatorname{tg}(-\frac{8\pi}{3}) = -\operatorname{tg}(\frac{8\pi}{3}) = -\operatorname{tg}(2\pi + \frac{2\pi}{3}) = -\operatorname{tg}(\frac{2\pi}{3}) = -(-\sqrt{3}) = \sqrt{3} \)

Ответ: 17) √3/3; 18) -√3; 19) -√3/3; 20) √3.