# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**The Cosine Theorem**

**From:**Herbert Prinz

**Date:**2002 Oct 19, 12:48 +0000

Hello Bill, Regiomontanus stated the cosine theorem for the spherical triangle for the first time in its general form in his "Five books about triangles of each kind" in 1533. Algebraic notation was not yet invented, so the theorem was expressed in plain language in form of a proportion. Not only is the proof rather hard to follow, even the statement itself is not easily recognizable as the law of cosines. Barnabas Hughes translates it in his book "Regiomontanus On Triangles", 1967, as follows: "In every spherical triangle that is constructed from the arcs of great circles, the ratio of the versed sine of any angle to the difference of two versed sines, of which one is [the versed sine] of the side subtending this angle while the other is [the versed sine] of the difference of the two arcs including this angle, is as the ratio of the square of the whole right sine to the rectangular product of the sines of the arcs placed around the mentioned angle." Since then we have come a long way. Using algebra, the cosine theorem for the spherical triangle is a trivial consequence of the equality of the scalar product of two unit vectors with the cosine of the angle between them. Convert two positions on the unit sphere given in spherical coordinates into rectangular coordinates of the corresponding position vectors (sometimes called "direction cosines"), multiply the vectors, and you are done. It's actually much more work to write it all down than to "see" it: Without sacrificing general validity, we may assume that one point is at longitude 0, because we can rotate them both together without changing their relative position. In that case, Lon_b represents the difference in longitude between the two points, i.e. their hour angle. We have a = (cos Lat_a, 0, sin Lat_a) b = (cos Lat_b * cos Lon_b, cos Lat_b * sin Lon_b, sin Lat_b) and therefore cos angle(a,b) = a . b = cos Lat_a * cos Lat_b * cos Lon_b + 0 + sin Lat_a * sin Lat_b That's really all. The beauty of Smart's purely geometric derivation is that it shows how all "spherical" trigonometry is in fact plane trigonometry in disguise. The law of cosines for spherical triangles is actually a formula about the relation of four angles in an irregular tetrahedron (see fig. 3 in his textbook). Three of those angles are between the three edges of the tetrahedron meeting in the center of the sphere and one is at the corner where the tangential plane touches the sphere. One can completely leave the sphere out of the game and still proof the law of cosines. This is one possible key to understanding why "spherical" trigonometry works for celestial navigation on a non-spherical Earth. It works, because the navigational triangle is about the mutual angles between three verticals, and not about distances on the surface of the Earth. WSMurdoch@AOL.COM wrote: > Chapter 1 section 5 derives the cosine formula from simple plane trig. > I have used it in USPS JN courses to answer the question, "Where did > that formula come from?" It is not too hard to pick your way through. > > Bill Murdoch