While solving a question, I met the next problem, for what x, is: $$ \frac{1}{\pi} \cdot \cos^{-1}(x) \in \mathbb{Q} $$
I found in this paper that for $ 0 \leq r \leq 1, r \in \mathbb{Q} $,
$$ \frac{1}{\pi} \cdot \cos^{-1}(\sqrt{r}) \in \mathbb{Q} \iff r = 0,\frac{1}{4},\frac{1}{2},\frac{3}{4}, 1$$
(the same holds for $\sin^{-1} $)
I would like to know if there is more known about the matter.
(for example what about $\frac{1}{\pi} \cdot\cos^{-1} \sqrt[3]{r}$)
The paper was written by Juan L. Varona