Задание 1. Решите простейшие тригонометрические уравнения a) 2cos( - ) = √3 6) sin(-) = √2 2 в) cos(-2x) = r) tg(-4x) = √3 д) ctg (-) = 1 √3 2 π 6 x 2 x 3 x 2

а)

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

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

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

Случай 1:

\[\frac{x}{2} = \frac{\pi}{6} + \frac{\pi}{6} + 2\pi k\]

\[\frac{x}{2} = \frac{\pi}{3} + 2\pi k\]

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

Случай 2:

\[\frac{x}{2} = -\frac{\pi}{6} + \frac{\pi}{6} + 2\pi k\]

\[\frac{x}{2} = 2\pi k\]

\[x = 4\pi k, k \in \mathbb{Z}\]

б)

\[\sin(-\frac{x}{3}) = \frac{\sqrt{2}}{2}\]

\[-\frac{x}{3} = (-1)^n \frac{\pi}{4} + \pi n, n \in \mathbb{Z}\]

\[x = -3((-1)^n \frac{\pi}{4} + \pi n), n \in \mathbb{Z}\]

\[x = (-1)^{n+1} \frac{3\pi}{4} - 3\pi n, n \in \mathbb{Z}\]

в)

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

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

\[2x = \pm \frac{5\pi}{6} + 2\pi k, k \in \mathbb{Z}\]

\[x = \pm \frac{5\pi}{12} + \pi k, k \in \mathbb{Z}\]

г)

\[\tan(-4x) = \sqrt{3}\]

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

\[x = -\frac{\pi}{12} - \frac{\pi}{4}k, k \in \mathbb{Z}\]

д)

\[\cot(-\frac{x}{2}) = 1\]

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

\[x = -\frac{\pi}{2} - 2\pi k, k \in \mathbb{Z}\]