What are the exact angles of the triangular faces of each of the rhombic pyramids of the icosahedron stellation, the compound of five octahedra?

Question

I'm talking about this compound of five regular octahedra. Based on looking at the icosahedron stellation diagram and making an educated guess about where the golden ratio comes in, I calculated the angles as$ $ $\theta, \frac{\pi}{3}$, and $\frac{2\pi}{3}-\theta$ where$ $ $\theta = \frac{\pi}{3} + 2 \arctan( \frac{\sqrt{5} - 2}{\sqrt{3}})$$, $ but am not confident I am right — thought it was weird it didnt reduce to a closed form solution.