cos((x/2)+(pi/3)) inequation

Given the inequality:
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 0$$
Solve:
Given the equation
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation

with the change of sign in 0

We get:
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{\pi}{6}$$
$$\frac{x}{2} = \pi n - \frac{5 \pi}{6}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{3}$$
$$x_{2} = 2 \pi n - \frac{5 \pi}{3}$$
$$x_{1} = 2 \pi n + \frac{\pi}{3}$$
$$x_{2} = 2 \pi n - \frac{5 \pi}{3}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{3}$$
$$x_{2} = 2 \pi n - \frac{5 \pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{3}$$
substitute to the expression
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} > 0$$
$$\cos{\left(\frac{2 \pi n - \frac{1}{10} + \frac{\pi}{3}}{2} + \frac{\pi}{3} \right)} > 0$$

-sin(-1/20 + pi*n) > 0

Then
$$x < 2 \pi n + \frac{\pi}{3}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n + \frac{\pi}{3} \wedge x < 2 \pi n - \frac{5 \pi}{3}$$

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