Given the equation
$$\cos{\left(x + \frac{\pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\cos{\left(x + \frac{\pi}{3} \right)} = 0$$
This equation is transformed to
$$x + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$x + \frac{\pi}{3} = \pi n + \frac{\pi}{2}$$
$$x + \frac{\pi}{3} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n + \frac{\pi}{6}$$
$$x = \pi n - \frac{5 \pi}{6}$$