Задание 2. Решите простейшие тригонометрические неравенства a) cos()> √3 2 1 6) sin (2x) <= 2 π в) 2 cos (2x +-) ≤ 1 3 г) 2 cos (4x -> √3 д)√3 tg(3x+) < 1 π 6 x 3

а)

\[\cos(\frac{x}{3}) > \frac{\sqrt{3}}{2}\]

\[-\frac{\pi}{6} + 2\pi k < \frac{x}{3} < \frac{\pi}{6} + 2\pi k, k \in \mathbb{Z}\]

\[-\frac{\pi}{2} + 6\pi k < x < \frac{\pi}{2} + 6\pi k, k \in \mathbb{Z}\]

б)

\[\sin(2x) \le \frac{1}{2}\]

\[-\frac{7\pi}{6} + 2\pi k \le 2x \le \frac{\pi}{6} + 2\pi k, k \in \mathbb{Z}\]

\[-\frac{7\pi}{12} + \pi k \le x \le \frac{\pi}{12} + \pi k, k \in \mathbb{Z}\]

в)

\[2 \cos(2x + \frac{\pi}{3}) \le 1\]

\[\cos(2x + \frac{\pi}{3}) \le \frac{1}{2}\]

\[\frac{\pi}{3} + 2\pi k \le 2x + \frac{\pi}{3} \le \frac{5\pi}{3} + 2\pi k, k \in \mathbb{Z}\]

\[\frac{2\pi}{3} + 2\pi k \le 2x \le \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z}\]

\[\frac{\pi}{3} + \pi k \le x \le \frac{2\pi}{3} + \pi k, k \in \mathbb{Z}\]

г)

\[2 \cos(4x - \frac{\pi}{6}) > \sqrt{3}\]

\[\cos(4x - \frac{\pi}{6}) > \frac{\sqrt{3}}{2}\]

\[-\frac{\pi}{6} + 2\pi k < 4x - \frac{\pi}{6} < \frac{\pi}{6} + 2\pi k, k \in \mathbb{Z}\]

\[2\pi k < 4x < \frac{\pi}{3} + 2\pi k, k \in \mathbb{Z}\]

\[\frac{\pi k}{2} < x < \frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}\]

д)

\[\sqrt{3} \tan(3x + \frac{\pi}{6}) < 1\]

\[\tan(3x + \frac{\pi}{6}) < \frac{1}{\sqrt{3}}\]

\[-\frac{\pi}{2} + \pi k < 3x + \frac{\pi}{6} < \frac{\pi}{6} + \pi k, k \in \mathbb{Z}\]

\[-\frac{2\pi}{3} + \pi k < 3x < \pi k, k \in \mathbb{Z}\]

\[-\frac{2\pi}{9} + \frac{\pi k}{3} < x < \frac{\pi k}{3}, k \in \mathbb{Z}\]