5. Найдите корни уравнения tg\left(3x+\frac{\pi}{6}\right)=1.

Решим уравнение $$tg(3x + \frac{\pi}{6}) = 1$$.

$$3x + \frac{\pi}{6} = arctg(1) + \pi n, n \in Z$$

$$3x + \frac{\pi}{6} = \frac{\pi}{4} + \pi n, n \in Z$$

$$3x = \frac{\pi}{4} - \frac{\pi}{6} + \pi n, n \in Z$$

$$3x = \frac{3\pi - 2\pi}{12} + \pi n, n \in Z$$

$$3x = \frac{\pi}{12} + \pi n, n \in Z$$

$$x = \frac{\pi}{36} + \frac{\pi n}{3}, n \in Z$$

Ответ: $$x = \frac{\pi}{36} + \frac{\pi n}{3}, n \in Z$$