(1/2)cos(x/3)>0 inequation

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

with the change of sign in 0

We get:
$$\frac{\cos{\left(\frac{x}{3} \right)}}{2} = 0$$

Divide both parts of the equation by 1/2

The equation is transformed to
$$\cos{\left(\frac{x}{3} \right)} = 0$$
This equation is transformed to
$$\frac{x}{3} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$\frac{x}{3} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$\frac{x}{3} = \pi n + \frac{\pi}{2}$$
$$\frac{x}{3} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{3}$$
$$x_{1} = 3 \pi n + \frac{3 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{3 \pi}{2}$$
$$x_{1} = 3 \pi n + \frac{3 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{3 \pi}{2}$$
This roots
$$x_{1} = 3 \pi n + \frac{3 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{3 \pi}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(3 \pi n + \frac{3 \pi}{2}\right) + - \frac{1}{10}$$
=
$$3 \pi n - \frac{1}{10} + \frac{3 \pi}{2}$$
substitute to the expression
$$\frac{\cos{\left(\frac{x}{3} \right)}}{2} > 0$$
$$\frac{\cos{\left(\frac{3 \pi n - \frac{1}{10} + \frac{3 \pi}{2}}{3} \right)}}{2} > 0$$

-sin(-1/30 + pi*n)     
------------------- > 0
         2             

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

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       x1      x2