Absolute convergence of $\sum_{n=1}^\infty n^{-5} \csc(2n\pi \sqrt[3]{3})$

Question

I want to prove the absolute convergence of this serie. There are two hints:

1. Firstly, prove that $|(\sin(\pi x)| \geq |x|$ for $|x| \leq \frac{1}{2}$.
2. Then use the fact that $2\sqrt[3]{3}$ is algebraic of degree 3.

It's easy to see the inequality of 1 if you draw both functions on the interval [-1/2,1/2], but there's a numerical way to show it? After proving this, we know that $\csc(x) = \frac{1}{\sin(x)}$, so can we conlude that $|\csc({\pi n})| \leq \frac{1}{|x|}$? I also know that $\sum_{n=1}^\infty |\frac{1}{n^5}|$ converges to $\zeta(5)$. I can't link these two points. Why do I have to use that $2\sqrt[3]{3}$ is algebraic of degree 3? Here's something that is not clear to me.

Any help will be useful to me!