# NavList:

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

**Re: Spherical Law of Cosines**

**From:**Dan Allen

**Date:**2002 Oct 23, 09:53 -0700

I spoke too hastily about my error in alternate renderings of the spherical law of cosines formula. Originally I had said that cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) was an alternate form, and then Bill Arden pointed out that for use with Hc it should have read sin(Hc) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) the difference being the left hand side of the equation. However, I went back and found support in Smart's book for the form that I had written, i.e., cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) in determining the length of twilight and other such calculations. In thinking about things I realized that both versions are right, but it simply is a matter of origin. Are the angles measured down from the pole (co-latitudes and such) or are they measured from the equator up (latitudes)? They are equivalent. The mental picture that I work from is the canonical version, cos(c) = cos(a)*cos(b) + sin(a)*sin(b)*cos(ab) and then I don't get into problems, because that is the one that I learned from. So we were both right. Dan