Simple question, but I don't know the answer and can't easily find good resources.
Sometimes we can give quite surprising exact forms to the circular functions at peculiar arguments through quite nontrivial methods, so I wasn't sure whether this was possible.
Are the non-right angles corresponding to Pythagorean triples transcendental multiples of $\pi$, or does it depend on the specific case?
Let's say as a simple example we take the Pythagorean triple $(3,4,5)$. One angle in this triangle would be $\tan^{-1}\frac34$.
Is this a transcendental multiple of $\pi$?
If so, or if not, can we generalise?
Edit: Original phrasing asked whether the angles were transcendental. This was just a mistake in typing due to tiredness, and attempts to correct this were also phrased badly due to more tiredness. Thanks for answering the question as intended rather than the literal interpretation.
They're transcendental (in radians). This follows from the Lindemann-Weierstrass theorem, which gives that if $\theta$ is algebraic then $z = e^{i \theta}$ is transcendental, and hence $\tan \theta = -i \frac{z - z^{-1}}{z + z^{-1}}$ is also transcendental. Taking the contrapositive, if $\tan \theta$ is algebraic then $\theta$ is transcendental. (This is a stronger result than what you asked for and in particular implies immediately that $\pi$ is transcendental.)
You may have meant to ask not whether $\theta$ is transcendental but whether $\theta$ is a transcendental multiple $\pi \alpha$ of $\pi$, though (equivalently you may have wanted to know about degrees, not radians). In that case the angles are still transcendental but now it follows from the Gelfond-Schneider theorem, applied to $e^{i \pi \alpha} = (-1)^{\alpha}$. We get that if $\alpha$ is algebraic but irrational then $z = e^{i \pi \alpha}$ is transcendental, and as above that $\tan \pi \alpha$ is transcendental. Taking the contrapositive again, if $\tan \pi \alpha$ is algebraic then $\alpha$ is either transcendental or rational. And it's a nice exercise to show that if $\alpha$ is rational then $\tan \pi \alpha$ is rational (or infinite) iff $\alpha$ is an integer multiple of $\frac{1}{4}$; the corresponding values of the tangent are $0, \pm 1, \pm \infty$ and this doesn't cover any Pythagorean triples.