We know all the fractions of $\pi$ for which $sin(\frac{k}{n} \pi)$ is rational thanks to Nievens theorem, same for cosine.
But I wonder if there is a similar theorem for the tangent $tan(\frac{k}{n} \pi)$?
Apart $tan(\frac{1}{4} \pi) = 1$ and $tan(0 \pi) = 0$, I fail to find any other case
The beautiful identities at https://mathworld.wolfram.com/Tangent.html all lead to algebraic irrational for small fractions of $\pi$. Is it a known result?
Forget it, it's a case of blindness.
It's covered by Nievens theorem, as reported on wikipedia page https://en.wikipedia.org/wiki/Niven%27s_theorem