# NavList:

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

**Spherical Law of Cosines**

**From:**Dan Allen

**Date:**2002 Oct 18, 10:05 -0700

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