sin(t)>1/3 inequation

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

sin(-1/10 + 2*pi*n + asin(1/3)) > 1/3

Then
$$t < 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}$$
no execute
one of the solutions of our inequality is:
$$t > 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)} \wedge t < 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$

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       t1      t2