MHB Find Angles of Spherical Triangle $\mathcal{P}$

AI Thread Summary
The discussion revolves around finding the angles of a spherical triangle defined by vertices at specific coordinates. The cosine of the angles at the vertices is provided, with one angle being zero and the others calculated as 1/sqrt(3). The cosine formula for spherical triangles is mentioned, but the user initially struggles to apply it effectively. Ultimately, the user indicates they have solved the problem, suggesting that the angles have been successfully determined. The conversation highlights the application of spherical trigonometry in calculating triangle angles.
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Consider the spherical triangle $\mathcal{P}$ with vertices $P_1 = (1,0,0)$, $P_2 = (0,1,0)$ and $P_3 = (1/\sqrt{3}, 1/\sqrt{3},1/\sqrt{3})$. Find the angles $\phi_1, \phi_2, \phi_3$ of $\mathcal{P}$ at $P_1, P_2, P_3$ respectively.

I know the cosine angles are $\cos(\theta_1) = 0$, $\cos(\theta_2) = \cos(\theta_3) = 1/{\sqrt{3}}$. I know the cosine formula is $\cos{c} = \cos{a}\cos{b}+\sin{a}\sin{b}\cos{C}.$ However, I can't put these ideas together to find the angles. Could someone please show me how to do that? Thanks.
 
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