Для решения тригонометрических уравнений будем использовать общие формулы:
### Столбец А
1. $\cos x = \frac{1}{2}$
$x = \pm \frac{\pi}{3} + 2\pi n, n \in \mathbb{Z}$
3. $\sin x = \frac{\sqrt{2}}{2}$
$x = (-1)^n \frac{\pi}{4} + \pi n, n \in \mathbb{Z}$
5. $\text{tg} x = -1$
$x = -\frac{\pi}{4} + \pi n, n \in \mathbb{Z}$
7. $\text{ctg} x = \frac{\sqrt{3}}{3}$
$x = \frac{\pi}{3} + \pi n, n \in \mathbb{Z}$
9. $\sin x = -1$
$x = -\frac{\pi}{2} + 2\pi n, n \in \mathbb{Z}$
11. $\text{tg} x = -\frac{\sqrt{3}}{3}$
$x = -\frac{\pi}{6} + \pi n, n \in \mathbb{Z}$
13. $\sin x = 0$
$x = \pi n, n \in \mathbb{Z}$
15. $\cos x = 1,5$
Корней нет, так как $|\cos x| \le 1$.
17. $\cos x = -\frac{3}{5}$
$x = \pm \arccos(-\frac{3}{5}) + 2\pi n = \pm (\pi - \arccos(\frac{3}{5})) + 2\pi n, n \in \mathbb{Z}$
19. $\cos x = 0,4$
$x = \pm \arccos(0,4) + 2\pi n, n \in \mathbb{Z}$
### Столбец B
1. $\text{tg}(2x - \frac{\pi}{6}) = \sqrt{3}$
$2x - \frac{\pi}{6} = \frac{\pi}{3} + \pi n \Rightarrow 2x = \frac{\pi}{2} + \pi n \Rightarrow x = \frac{\pi}{4} + \frac{\pi n}{2}, n \in \mathbb{Z}$
3. $\cos(3x + \frac{\pi}{3}) = \frac{1}{2}$
$3x + \frac{\pi}{3} = \pm \frac{\pi}{3} + 2\pi n$
$3x_1 = \frac{\pi}{3} - \frac{\pi}{3} + 2\pi n = 2\pi n \Rightarrow x_1 = \frac{2\pi n}{3}$
$3x_2 = -\frac{\pi}{3} - \frac{\pi}{3} + 2\pi n = -\frac{2\pi}{3} + 2\pi n \Rightarrow x_2 = -\frac{2\pi}{9} + \frac{2\pi n}{3}, n \in \mathbb{Z}$
5. $\cos(2x + \frac{\pi}{4}) = 0$
$2x + \frac{\pi}{4} = \frac{\pi}{2} + \pi n \Rightarrow 2x = \frac{\pi}{4} + \pi n \Rightarrow x = \frac{\pi}{8} + \frac{\pi n}{2}, n \in \mathbb{Z}$
7. $\text{tg}(\frac{x}{2} - \frac{\pi}{2}) = -\sqrt{3}$
$\frac{x}{2} - \frac{\pi}{2} = -\frac{\pi}{3} + \pi n \Rightarrow \frac{x}{2} = \frac{\pi}{6} + \pi n \Rightarrow x = \frac{\pi}{3} + 2\pi n, n \in \mathbb{Z}$
9. $\text{ctg}(5x + \frac{\pi}{4}) = \frac{\sqrt{3}}{3}$
$5x + \frac{\pi}{4} = \frac{\pi}{3} + \pi n \Rightarrow 5x = \frac{\pi}{12} + \pi n \Rightarrow x = \frac{\pi}{60} + \frac{\pi n}{5}, n \in \mathbb{Z}$
11. $\sin(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
$x - \frac{\pi}{4} = (-1)^n \frac{\pi}{4} + \pi n$
$x = \frac{\pi}{4} + (-1)^n \frac{\pi}{4} + \pi n$
При четном $n=2k$: $x = \frac{\pi}{4} + \frac{\pi}{4} + 2\pi k = \frac{\pi}{2} + 2\pi k$
При нечетном $n=2k+1$: $x = \frac{\pi}{4} - \frac{\pi}{4} + \pi(2k+1) = \pi + 2\pi k$
$x = \frac{\pi}{2} + 2\pi k; x = \pi + 2\pi k, k \in \mathbb{Z}$
13. $\sin(\frac{\pi}{6} - x) = \frac{\sqrt{3}}{2}$
$\frac{\pi}{6} - x = (-1)^n \frac{\pi}{3} + \pi n \Rightarrow x = \frac{\pi}{6} - (-1)^n \frac{\pi}{3} - \pi n, n \in \mathbb{Z}$
15. $\cos(\frac{3\pi}{2} + x) = \frac{\sqrt{3}}{2}$
$\sin x = \frac{\sqrt{3}}{2}$ (так как $\cos(\frac{3\pi}{2} + x) = \sin x$)
$x = (-1)^n \frac{\pi}{3} + \pi n, n \in \mathbb{Z}$
17. $\sin(x + \frac{\pi}{6}) = 1$
$x + \frac{\pi}{6} = \frac{\pi}{2} + 2\pi n \Rightarrow x = \frac{\pi}{3} + 2\pi n, n \in \mathbb{Z}$
19. $\text{tg}(x - \frac{\pi}{4}) = 1$
$x - \frac{\pi}{4} = \frac{\pi}{4} + \pi n \Rightarrow x = \frac{\pi}{2} + \pi n, n \in \mathbb{Z}$
