1. Решите уравнения: a) cos(5x + \frac{\pi}{6}) = \frac{1}{2} б) sin(\frac{x}{4} - \frac{\pi}{6}) = -\frac{\sqrt{3}}{2} в) tg(3x - \frac{\pi}{5}) = -\frac{1}{\sqrt{3}} г) cos3x = -1.6 д) sin(5x - \frac{\pi}{3}) = 1 e) tg(6x - \frac{\pi}{3}) = 0 ж) -2ctg²x - 3ctgx + 2 = 0 з) 2sin²x - 3sinxcosx = 0 2. Решите неравенства a) sin4x≤ -\frac{1}{2} б) cos(\frac{x}{9}) ≤ -\frac{\sqrt{2}}{2}

1. Решите уравнения:

1. a) cos(5x + \(\frac{\pi}{6}\)) = \(\frac{1}{2}\)
• \(5x + \frac{\pi}{6} = \pm \frac{\pi}{3} + 2\pi k, k \in \mathbb{Z}\)
• Разделим на два случая:
• • \(5x + \frac{\pi}{6} = \frac{\pi}{3} + 2\pi k\)
  • \(5x = \frac{\pi}{3} - \frac{\pi}{6} + 2\pi k\)
  • \(5x = \frac{\pi}{6} + 2\pi k\)
  • \(x = \frac{\pi}{30} + \frac{2\pi k}{5}, k \in \mathbb{Z}\)
• • \(5x + \frac{\pi}{6} = -\frac{\pi}{3} + 2\pi k\)
  • \(5x = -\frac{\pi}{3} - \frac{\pi}{6} + 2\pi k\)
  • \(5x = -\frac{\pi}{2} + 2\pi k\)
  • \(x = -\frac{\pi}{10} + \frac{2\pi k}{5}, k \in \mathbb{Z}\)
2. б) sin(\(\frac{x}{4} - \frac{\pi}{6}\)) = -\(\frac{\sqrt{3}}{2}\)
• \(\frac{x}{4} - \frac{\pi}{6} = (-1)^k \left(-\frac{\pi}{3}\right) + \pi k, k \in \mathbb{Z}\)
• Разделим на два случая:
• • \(\frac{x}{4} - \frac{\pi}{6} = -\frac{\pi}{3} + 2\pi k\)
  • \(\frac{x}{4} = -\frac{\pi}{3} + \frac{\pi}{6} + 2\pi k\)
  • \(\frac{x}{4} = -\frac{\pi}{6} + 2\pi k\)
  • \(x = -\frac{2\pi}{3} + 8\pi k, k \in \mathbb{Z}\)
• • \(\frac{x}{4} - \frac{\pi}{6} = \pi + \frac{\pi}{3} + 2\pi k\)
  • \(\frac{x}{4} = \frac{4\pi}{3} + \frac{\pi}{6} + 2\pi k\)
  • \(\frac{x}{4} = \frac{9\pi}{6} + 2\pi k\)
  • \(x = 6\pi + 8\pi k, k \in \mathbb{Z}\)
3. в) tg(3x - \(\frac{\pi}{5}\)) = -\(\frac{1}{\sqrt{3}}\)
• \(3x - \frac{\pi}{5} = -\frac{\pi}{6} + \pi k, k \in \mathbb{Z}\)
• \(3x = \frac{\pi}{5} - \frac{\pi}{6} + \pi k\)
• \(3x = \frac{\pi}{30} + \pi k\)
• \(x = \frac{\pi}{90} + \frac{\pi k}{3}, k \in \mathbb{Z}\)
4. г) cos3x = -1.6
• Так как \(|cos3x| \le 1\), то уравнение не имеет решений.
5. д) sin(5x - \(\frac{\pi}{3}\)) = 1
• \(5x - \frac{\pi}{3} = \frac{\pi}{2} + 2\pi k, k \in \mathbb{Z}\)
• \(5x = \frac{\pi}{2} + \frac{\pi}{3} + 2\pi k\)
• \(5x = \frac{5\pi}{6} + 2\pi k\)
• \(x = \frac{\pi}{6} + \frac{2\pi k}{5}, k \in \mathbb{Z}\)
6. e) tg(6x - \(\frac{\pi}{3}\)) = 0
• \(6x - \frac{\pi}{3} = \pi k, k \in \mathbb{Z}\)
• \(6x = \frac{\pi}{3} + \pi k\)
• \(x = \frac{\pi}{18} + \frac{\pi k}{6}, k \in \mathbb{Z}\)
7. ж) -2ctg²x - 3ctgx + 2 = 0
• Замена: \(t = ctgx\)
• \(-2t^2 - 3t + 2 = 0\)
• \(2t^2 + 3t - 2 = 0\)
• \(D = 3^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25\)
• \(t_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}\)
• \(t_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2\)
• \(ctgx = \frac{1}{2}\) или \(ctgx = -2\)
• \(x = arctg(2) + \pi k, k \in \mathbb{Z}\) или \(x = arctg(-\frac{1}{2}) + \pi k, k \in \mathbb{Z}\)
8. з) 2sin²x - 3sinxcosx = 0
• \(sinx(2sinx - 3cosx) = 0\)
• \(sinx = 0\) или \(2sinx - 3cosx = 0\)
• \(x = \pi k, k \in \mathbb{Z}\) или \(2sinx = 3cosx\)
• \(tgx = \frac{3}{2}\)
• \(x = arctg(\frac{3}{2}) + \pi k, k \in \mathbb{Z}\)

2. Решите неравенства

1. a) sin4x ≤ -\(\frac{1}{2}\)
• \(-\frac{5\pi}{6} + 2\pi k \le 4x \le -\frac{\pi}{6} + 2\pi k, k \in \mathbb{Z}\)
• \(-\frac{5\pi}{24} + \frac{\pi k}{2} \le x \le -\frac{\pi}{24} + \frac{\pi k}{2}, k \in \mathbb{Z}\)
2. б) cos(\(\frac{x}{9}\)) ≤ -\(\frac{\sqrt{2}}{2}\)
• \(\frac{3\pi}{4} + 2\pi k \le \frac{x}{9} \le \frac{5\pi}{4} + 2\pi k, k \in \mathbb{Z}\)
• \(\frac{27\pi}{4} + 18\pi k \le x \le \frac{45\pi}{4} + 18\pi k, k \in \mathbb{Z}\)

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