sint<0 inequation

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

with the change of sign in 0

We get:
$$\sin{\left(t \right)} = 0$$
This equation is transformed to
$$t = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$t = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$t = 2 \pi n$$
$$t = 2 \pi n + \pi$$
, where n - is a integer
$$t_{1} = 2 \pi n$$
$$t_{2} = 2 \pi n + \pi$$
$$t_{1} = 2 \pi n$$
$$t_{2} = 2 \pi n + \pi$$
This roots
$$t_{1} = 2 \pi n$$
$$t_{2} = 2 \pi n + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$t_{0} < t_{1}$$
For example, let's take the point
$$t_{0} = t_{1} - \frac{1}{10}$$
=
$$2 \pi n + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(t \right)} < 0$$
$$\sin{\left(2 \pi n - \frac{1}{10} \right)} < 0$$

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

one of the solutions of our inequality is:
$$t < 2 \pi n$$

 _____           _____          
      \         /
-------ο-------ο-------
       t1      t2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$t < 2 \pi n$$
$$t > 2 \pi n + \pi$$