Though not really a “green” question, I happen to have been a math minor, so here goes:
The law of sines applies to all true two-dimensional triangles (triangles that are projected on a non-two-dimensional surface, like the earth, are not true triangles).
For example, take the triangle of angle pi/6 radians (30 degrees) on the unit cirlce. We know that the hypoteneuse is 1 (because it’s a unit circle), and that the lengths of the sides are half the square root of three and half the square root of 1 (or 1/2). The law of sines says that:
(Sin A)/a = (Sin B)/b = (Sin C)/c
or a general equality of proportions. If the hypoteneuse is a, the opposite side b, and the adjacent side c, then we have:
(Sin 90)/1 = (Sin 30)/(1/2) = (Sin 60)/(.5*sqrt(3))
All are equal to one.
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