sin(x)<=0 inequation

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

with the change of sign in 0

We get:
$$\sin{\left(x \right)} = 0$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
This roots
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$2 \pi n + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \leq 0$$
$$\sin{\left(2 \pi n - \frac{1}{10} \right)} \leq 0$$

sin(-1/10 + 2*pi*n) <= 0

one of the solutions of our inequality is:
$$x \leq 2 \pi n$$

 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n$$
$$x \geq 2 \pi n + \pi$$