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

In summary, a spherical triangle is a triangle formed on the surface of a sphere by three intersecting great circles. It is important to find the angles of a spherical triangle in various fields such as geography, astronomy, and navigation. The angles can be found using trigonometric formulas that take into account the curvature of the sphere. Real-world applications of finding the angles include calculating the position of stars, determining the shortest distance between two points, and creating accurate maps. There are also special cases, such as right spherical triangles and isosceles spherical triangles, which have specific formulas for finding the 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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I've solved this now!
 

1. What is a spherical triangle?

A spherical triangle is a triangle formed on the surface of a sphere by three intersecting great circles. It is analogous to a standard triangle on a flat plane.

2. Why is it important to find the angles of a spherical triangle?

Finding the angles of a spherical triangle is important in various fields such as geography, astronomy, and navigation. It allows us to determine the positions of objects on the Earth's surface or in space.

3. How do you find the angles of a spherical triangle?

The angles of a spherical triangle can be found using various trigonometric formulas, such as the law of cosines and the law of sines. These formulas take into account the curvature of the sphere and allow us to calculate the angles accurately.

4. What are some real-world applications of finding the angles of a spherical triangle?

Some examples of real-world applications include calculating the position of stars in astronomy, determining the shortest distance between two points on the Earth's surface in navigation, and creating accurate maps in geography.

5. Are there any special cases when finding the angles of a spherical triangle?

Yes, there are special cases when finding the angles of a spherical triangle. These include right spherical triangles, where one angle is a right angle, and isosceles spherical triangles, where two sides are of equal length. These cases have specific formulas for finding the angles.

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