# NavList:

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

**Re: Spherical Law of Cosines**

**From:**Dave Weilacher

**Date:**2002 Oct 18, 10:12 -0700

Thanks. Very pleased with you. Dave On Fri, 18 Oct 2002 10:05:29 -0700 Dan Allenwrote: > On Friday, October 18, 2002, at 07:53 AM, David > Weilacher wrote: > > > Please define little (a), (b), (c), and (ab), > for me. > > With pleasure! > > cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * > cos(ab) > > a is the length of the first side of the > spherical triangle. > b is the length of the second side of the > spherical triangle. > ab is the angle between the two sides a and b. > c is the length of the side opposite the angle > ab. > > Here is an example of a great circle distance > between > San Francisco (SF) and Salt Lake City (SLC). > All angles > and lengths are expressed in degrees for this > example, > and North and West are positive. > > SF: > lat1 = 37 degrees > lon1 = 122 degrees > > SLC: > lat2 = 40 degrees > lon2 = 112 degrees > > So, > > a is the co-latitude of lat1, or 90-lat1 or 53 > degrees. > b is the co-latitude of lat2, or 90-lat2 or 50 > degrees. > ab is the difference of the longitudes, or > lon1-lon2 or 10 degrees. > > Solving for c one learns the distance in > degrees between SF and SLC, > which is about 8.375 degrees. To get a > distance we understand, > multiply degrees times 60 nmi per deg and you > get 502.5 nmi. > > One then can re-substitute back in and use the > formula again to > determine the initial great circle course. In > this case we > use a as co-latitude of lat1, b is the distance > we just > computed (8.375 degrees -- make sure to use > degrees), c is > the co-latitude of lat2, and the angle ab is > the initial > course. We have rotated the whole spherical > triangle around > to put the sides a,b, and c where we know their > values, > leaving ab to be solved for. Substituting and > solving one > determines that the initial course is 65.9588 > degrees. > > --- > > In sight reduction: > > a is the co-latitude of the assumed position > b is the declination of the body (say the sun) > ab is the hour angle of the body > c is the altitude of the body > > --- > > Note too that there are equivalent ways of > writing the spherical > law of cosines. One can write it as > > cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) > > where the pairs of sin/cos of a/b are switched. > This allows one to do > great circle calculations using latitudes > directly, > without using co-latitudes. The origin is > moved from the pole to > the equator, so to speak. This form is often > handier but the first > version is the canonical version. > > Hope this helps. > > Dan > Dave Weilacher .US Coast Guard licensed captain . #889968 .ASA certified sailing and celestial . navigation instructor #990800 .IBM AS400 RPG contract programmer