tgt<0 inequation

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

with the change of sign in 0

We get:
$$\tan{\left(t \right)} = 0$$
This equation is transformed to
$$t = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$t = \pi n$$
, where n - is a integer
$$t_{1} = \pi n$$
$$t_{1} = \pi n$$
This roots
$$t_{1} = \pi n$$
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
$$t_{0} = t_{1} - \frac{1}{10}$$
=
$$\pi n + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(t \right)} < 0$$
$$\tan{\left(\pi n - \frac{1}{10} \right)} < 0$$

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

the solution of our inequality is:
$$t < \pi n$$

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