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One way to calculate distance (separation angle) on a sphere is to use
vectors and the atan2 function or rectangular to polar conversion. Set x
to the dot product of the two vectors, and y to the magnitude of their
cross product. The separation angle is atan2(y, x), or you can get the
angle by converting x and y to polar form.

For example, Meeus calculates the separation angle between two stars:

14h 15m 39.7s +19°10′57″
13h 25m 11.6s -11°09′41″

He uses spherical trig. Using the vector method, start by forming the
rectanglar coordinates on the celestial sphere:

-.783786 -.526988 .328578
-.914079 -.356354 -.193573

dot product = .840633 (x)
cross product magnitude = .541606 (y)

Converting (x, y) to polar form gives angle 32.7930°, which agrees with
Meeus.

An advantage of the vector method is its accuracy at all angles. A large
portion of Meeus' chapter on separation angle discusses ways to avoid
the inaccuracy of the traditional spherical trig formula near 0 or 180
degrees.

   

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