public service math lesson: spherical trigonometry

So I've arbitrarily decided that more people need to know about spherical trigonometry. (e.g., just in case the GPS gets destroyed and we're stuck having to do our own navigation again.)

It's really the same solving of triangles that you learned to do in high school geometry/trig, i.e., gimme side-angle-side or angle-side-angle to nail down what the triangle actually is, then use Law of Sines or Law of Cosines or some combination thereof to work out the previously unknown sides/angles that you care about.

It's just that some of the rules for spherical trig are a Little Bit Different.

So, jumping right into the deep end, here's an application inspired by recent events:

You, an observer stationed on planet Earth at some particular latitude, want to know where the sun is going to be in the sky at some particular time of day, some particular day of the year.

And, cutting to the chase, here's a triangle to solve:

If you know b, A, c (side, angle, side), you can solve for a (opposite side) using the Law of Cosines

cos a = cos b cos c + sin b sin c cos A

and then B (next angle) using the Law of Sines

sinB/sinb = sinA/sina = sinC/sinc

( ...math wanking continues... )