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How to Compute the Distance Between Two Stars

Question:
How do you compute the distance between any two given stars?
Please illustrate by showing the calculations for the distance between Betelgeuse and Alnitak.
Answer:
This is actually an application of spherical trigonometry. A good description of this calculation, including an online tool to do the calculation for you using two methods (spherical trigonometry and stellar magnitude), can be found here.