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

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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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Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?

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