With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$.
Is it possible to approximate all angles with certain pythagorean triples?
And given an $\alpha$ and an $\epsilon$, how to get the example with smallest numbers $a,b,c$, such that $\alpha\pm \epsilon=\arccos \frac ac$?
A Pythagorean triangle has sides $2mn$, $m^2-n^2$, $m^2+n^2$ for some integers $m\gt n\gt0$. One of its angles has cosine equal to $${m^2-n^2\over m^2+n^2}=1-2{1\over(m/n)^2+1}$$ Now $m/n$ is dense in $(1,\infty)$, so that cosine is dense in $(0,1)$.