17.8 a) tgx = 1, 6) tg x = -√3/3; 17.9 a) tgx = 0; 6) tgx = -2; 17.10 a) ctgx = 1; 6) ctg x = -√3; 17.11 a) tg²x - 6tgx + 5 = 0; 6) tg²x - 2tgx - 3 = 0. • 17.12 a) tg (π + x) = √3; 6) 2 ctg (2n + x) - tg (π/2 + x) = √3; B) -√3 tg (π - x) = 1; r) ctg (2π – x) + tg (3π/2 + x) = 2.

Решаю тригонометрические уравнения.

1. 17.8 a) $$tg x = 1$$

$$x = arctg(1) + \pi n, n \in Z$$

$$x = \frac{\pi}{4} + \pi n, n \in Z$$

Ответ: $$x = \frac{\pi}{4} + \pi n, n \in Z$$

2. 17.8 б) $$tg x = -\frac{\sqrt{3}}{3}$$

$$x = arctg(-\frac{\sqrt{3}}{3}) + \pi n, n \in Z$$

$$x = -\frac{\pi}{6} + \pi n, n \in Z$$

Ответ: $$x = -\frac{\pi}{6} + \pi n, n \in Z$$

3. 17.9 a) $$tg x = 0$$

$$x = arctg(0) + \pi n, n \in Z$$

$$x = \pi n, n \in Z$$

Ответ: $$x = \pi n, n \in Z$$

4. 17.9 б) $$tg x = -2$$

$$x = arctg(-2) + \pi n, n \in Z$$

$$x = -arctg(2) + \pi n, n \in Z$$

Ответ: $$x = -arctg(2) + \pi n, n \in Z$$

5. 17.10 a) $$ctg x = 1$$

$$tg x = 1$$

$$x = arctg(1) + \pi n, n \in Z$$

$$x = \frac{\pi}{4} + \pi n, n \in Z$$

Ответ: $$x = \frac{\pi}{4} + \pi n, n \in Z$$

6. 17.10 б) $$ctg x = -\sqrt{3}$$

$$tg x = -\frac{\sqrt{3}}{3}$$

$$x = arctg(-\frac{\sqrt{3}}{3}) + \pi n, n \in Z$$

$$x = -\frac{\pi}{6} + \pi n, n \in Z$$

Ответ: $$x = -\frac{\pi}{6} + \pi n, n \in Z$$

7. 17.11 a) $$tg^2 x - 6tg x + 5 = 0$$

Пусть $$t = tg x$$, тогда

$$t^2 - 6t + 5 = 0$$

$$D = (-6)^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16$$

$$t_1 = \frac{6 + \sqrt{16}}{2 \cdot 1} = \frac{6 + 4}{2} = \frac{10}{2} = 5$$

$$t_2 = \frac{6 - \sqrt{16}}{2 \cdot 1} = \frac{6 - 4}{2} = \frac{2}{2} = 1$$

$$tg x = 5$$

$$x = arctg(5) + \pi n, n \in Z$$

$$tg x = 1$$

$$x = arctg(1) + \pi n, n \in Z$$

$$x = \frac{\pi}{4} + \pi n, n \in Z$$

Ответ: $$x = arctg(5) + \pi n, n \in Z$$; $$x = \frac{\pi}{4} + \pi n, n \in Z$$

8. 17.11 б) $$tg^2 x - 2tg x - 3 = 0$$

Пусть $$t = tg x$$, тогда

$$t^2 - 2t - 3 = 0$$

$$D = (-2)^2 - 4 \cdot 1 \cdot (-3) = 4 + 12 = 16$$

$$t_1 = \frac{2 + \sqrt{16}}{2 \cdot 1} = \frac{2 + 4}{2} = \frac{6}{2} = 3$$

$$t_2 = \frac{2 - \sqrt{16}}{2 \cdot 1} = \frac{2 - 4}{2} = \frac{-2}{2} = -1$$

$$tg x = 3$$

$$x = arctg(3) + \pi n, n \in Z$$

$$tg x = -1$$

$$x = arctg(-1) + \pi n, n \in Z$$

$$x = -\frac{\pi}{4} + \pi n, n \in Z$$

Ответ: $$x = arctg(3) + \pi n, n \in Z$$; $$x = -\frac{\pi}{4} + \pi n, n \in Z$$

9. 17.12 a) $$tg (\pi + x) = \sqrt{3}$$

$$tg (\pi + x) = tg x$$

$$tg x = \sqrt{3}$$

$$x = arctg(\sqrt{3}) + \pi n, n \in Z$$

$$x = \frac{\pi}{3} + \pi n, n \in Z$$

Ответ: $$x = \frac{\pi}{3} + \pi n, n \in Z$$

10. 17.12 б) $$2 ctg (2\pi + x) - tg (\frac{\pi}{2} + x) = \sqrt{3}$$

$$2 ctg x + ctg x = \sqrt{3}$$

$$3 ctg x = \sqrt{3}$$

$$ctg x = \frac{\sqrt{3}}{3}$$

$$tg x = \sqrt{3}$$

$$x = arctg(\sqrt{3}) + \pi n, n \in Z$$

$$x = \frac{\pi}{3} + \pi n, n \in Z$$

Ответ: $$x = \frac{\pi}{3} + \pi n, n \in Z$$

11. 17.12 в) $$\- \sqrt{3} tg (\pi - x) = 1$$

$$-\sqrt{3} \cdot (-tg x) = 1$$

$$\sqrt{3} tg x = 1$$

$$tg x = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

$$x = arctg(\frac{\sqrt{3}}{3}) + \pi n, n \in Z$$

$$x = \frac{\pi}{6} + \pi n, n \in Z$$

Ответ: $$x = \frac{\pi}{6} + \pi n, n \in Z$$

12. 17.12 г) $$ctg (2\pi - x) + tg (\frac{3\pi}{2} + x) = 2$$

$$-ctg x - ctg x = 2$$

$$-2 ctg x = 2$$

$$ctg x = -1$$

$$tg x = -1$$

$$x = arctg(-1) + \pi n, n \in Z$$

$$x = -\frac{\pi}{4} + \pi n, n \in Z$$

Ответ: $$x = -\frac{\pi}{4} + \pi n, n \in Z$$