Is $\frac{5\pi}{6}$ a transcendental or an algebraic number?

Question

$\Phi^2$ is an algebraic number as it is the root of $x^2-3x+1=0$

So knowing that $\frac{\pi}{6} = \frac{\Phi^2}{5}$, which is a relation I saw, does it mean that $\frac{5\pi}{6}$ is algebraic despite $\pi$ being transcendental?

How would you prove that it is or it is not?

Answer 1

Very easily. If $\frac56 \pi=a$, where $a$ is algebraic, then $\pi=\frac65 a$... This implies $\pi$ is algebraic...

Note: the product of algebraic numbers is algebraic...