Reciproc of the Lindemann theorem and the arc cosine of the golden ratio Via the Lindemann theorem it is easy to prove that the cosine of any rational multiple of $\pi$ is an algebraic number; however its contrapositive only tells us that the arc cosine of an algebraic number is a transcendental number, not necessarily linearly dependent from $\pi$ over the rationals.
I ask, then, is it true via any other theorem? Is there a nice counterexample?
I came to this question specifically as I've found in my research an angle $\theta$ such that $$\cos\theta = \frac{\sqrt{5}-1}{2} =: \varphi^{-1},$$ and I wanted to find out if $\theta$ has any nice representation, such as a rational multiple of $\pi$.

No. As a concrete counterexample, $\cos^{-1}(0.6)$ is not a rational multiple of $\pi$.
$\cos^{-1}(0.6)$ is a rational multiple of $\pi$ if and only if $0.6+0.8i$ is a root of unity. But if $(0.6+0.8i)^n = 1$ then $(3+4i)^n$ must be real. And this is impossible because $(3+4i)^n \equiv 3+4i \pmod 5$ for all $n\ge 1$
