# NavList:

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

**Re: Tables vs. Calculators**

**From:**Richard Irvine

**Date:**2002 Sep 23, 11:03 +0200

Since you mention both series expansions and the spherical cosine rule I thought I would just point out how this formula contains (at least) three for the price of one. If the sides of the spherical triangle are small then we can subsititute the first terms from the series expansion of sine and cosine For small x, sin x ~= x, cos x ~= 1 - x**2/2 (1 minus x squared on 2) Substituting into the spherical cosine formula > > cos(a)*cos(b) + sin(a)*sin(b)*cos(ab) = cos(c) > gives (1-a**2/2)(1-b**2/2) + abcosC = (1-c**2/2) (I have written C instead of your ab for the angle opposite the side c). or, rearranging, c**2 = a**2 + b**2 -2abcosC neglecting the term in a**2.b**2 (second order of small quantities) If we keep the sides of the triangle constant but inflate the sphere until its radius is so great that the spherical triangle is planar, then the approximation becomes exact. In other words this is a derivation of the planar cosine rule. This is how I check that I have remembered the spherical cosine rule correctly (having had the planar cosine rule hammered home at an age when my memory was more supple). The special case that C = 90 degrees then gives Pythagoras theorem. > -----Original Message----- > From: Dan Allen [mailto:danallen46{at}ATTBI.COM] > Sent: Friday, September 20, 2002 11:52 PM > To: NAVIGATION-L{at}listserv.webkahuna.com > Subject: Re: Tables vs. Calculators > > > On Friday, September 20, 2002, at 11:32 AM, Chuck Taylor wrote: > > > I could even reproduce the sines and cosines if I wanted to trouble > > myself with going through a Taylor series expansion. > > You're lucky: you've got a series expansion named after yourself, > but the rest of us (except for Maclaurins) are just out of luck... ;-) >> > Seriously, I totally agree with Chuck. I can and do have the basic > formula for great circle nav and sight reduction memorized because it > is so simple: > > cos(a)*cos(b) + sin(a)*sin(b)*cos(ab) = cos(c) > > Many of our nav problems boil down to using this simple formula. > > For example, for great circle problems with arguments in degrees > > IF a = 90 - lat1 > & b = 90 - Lat2 > & ab = Lon2 - Lon1 > THEN > c = GC Distance in degrees, or multiply it by 60 for > nautical miles > > or for sight reduction with arguments in degrees > > IF a = 90 - estimated latitude > & b = 90 - declination > & ab = LHA = GHA - estimated longitude > THEN > 90 - c = altitude > > This formula is easy to program into calculators, or easy to write > down on a piece of paper and do by hand with a basic scientific > calculator. This is the essence of self-reliant navigation. > > Dan >