sinx<=√2\2 inequation

Given the inequality:
$$\sin{\left(x \right)} \leq \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$x = 2 \pi n + \frac{\pi}{4}$$
$$x = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
$$x_{1} = 2 \pi n + \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \frac{3 \pi}{4}$$
$$x_{1} = 2 \pi n + \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \frac{3 \pi}{4}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \frac{3 \pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\sin{\left(x \right)} \leq \frac{\sqrt{2}}{2}$$
$$\sin{\left(2 \pi n - \frac{1}{10} + \frac{\pi}{4} \right)} \leq \frac{\sqrt{2}}{2}$$

                             ___
   /  1    pi         \    \/ 2 
sin|- -- + -- + 2*pi*n| <= -----
   \  10   4          /      2  
                           

one of the solutions of our inequality is:
$$x \leq 2 \pi n + \frac{\pi}{4}$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n + \frac{\pi}{4}$$
$$x \geq 2 \pi n + \frac{3 \pi}{4}$$