223. Решите уравнение:

Привет! Давай решим эти тригонометрические уравнения. Везде используем формулы общего решения для $\cos x = a$: 1) $\cos 6x = 1$ $6x = 2\pi n, n \in \mathbb{Z}$ $x = \frac{\pi n}{3}, n \in \mathbb{Z}$ 2) $\cos 4x = -\frac{\sqrt{3}}{2}$ $4x = \pm \arccos(-\frac{\sqrt{3}}{2}) + 2\pi n$ $4x = \pm \frac{5\pi}{6} + 2\pi n$ $x = \pm \frac{5\pi}{24} + \frac{\pi n}{2}, n \in \mathbb{Z}$ 3) $\cos \frac{9x}{5} = 0$ $\frac{9x}{5} = \frac{\pi}{2} + \pi n$ $x = \frac{5}{9} (\frac{\pi}{2} + \pi n) = \frac{5\pi}{18} + \frac{5\pi n}{9}, n \in \mathbb{Z}$ 4) $\cos(7x - \frac{\pi}{4}) = -1$ $7x - \frac{\pi}{4} = \pi + 2\pi n$ $7x = \frac{5\pi}{4} + 2\pi n$ $x = \frac{5\pi}{28} + \frac{2\pi n}{7}, n \in \mathbb{Z}$ 5) $\cos(2 - 5x) = \frac{1}{2}$ $2 - 5x = \pm \frac{\pi}{3} + 2\pi n$ $-5x = -2 \pm \frac{\pi}{3} + 2\pi n$ $x = \frac{2}{5} \mp \frac{\pi}{15} - \frac{2\pi n}{5}, n \in \mathbb{Z}$ 6) $\cos \frac{3\pi x}{4} = \frac{\sqrt{3}}{2}$ $\frac{3\pi x}{4} = \pm \frac{\pi}{6} + 2\pi n$ $x = \frac{4}{3\pi} (\pm \frac{\pi}{6} + 2\pi n) = \pm \frac{2}{9} + \frac{8n}{3}, n \in \mathbb{Z}$ 7) $\cos(2x + \frac{\pi}{6}) = -\frac{\pi}{3}$ Так как $|-\frac{\pi}{3}| \approx 1.05 > 1$, уравнение решений не имеет. 8) $\cos(6x - 5) = \frac{\pi}{8}$ $6x - 5 = \pm \arccos(\frac{\pi}{8}) + 2\pi n$ $x = \frac{5 \pm \arccos(\frac{\pi}{8}) + 2\pi n}{6}, n \in \mathbb{Z}$ 9) $2\cos(7x - \frac{\pi}{8}) + \sqrt{2} = 0$ $2\cos(7x - \frac{\pi}{8}) = -\sqrt{2}$ $\cos(7x - \frac{\pi}{8}) = -\frac{\sqrt{2}}{2}$ $7x - \frac{\pi}{8} = \pm \frac{3\pi}{4} + 2\pi n$ $7x = \frac{\pi}{8} \pm \frac{6\pi}{8} + 2\pi n$ $x = \frac{\pi \pm 6\pi + 16\pi n}{56}$ $x_1 = \frac{7\pi + 16\pi n}{56} = \frac{\pi}{8} + \frac{2\pi n}{7}$, $x_2 = \frac{-5\pi + 16\pi n}{56}, n \in \mathbb{Z}$ 10) $3\cos(2x - \frac{\pi}{3}) - 1 = 0$ $3\cos(2x - \frac{\pi}{3}) = 1$ $\cos(2x - \frac{\pi}{3}) = \frac{1}{3}$ $2x - \frac{\pi}{3} = \pm \arccos(\frac{1}{3}) + 2\pi n$ $2x = \frac{\pi}{3} \pm \arccos(\frac{1}{3}) + 2\pi n$ $x = \frac{\pi}{6} \pm \frac{1}{2}\arccos(\frac{1}{3}) + \pi n, n \in \mathbb{Z}$