# NavList:

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

**Re: Spherical Law of Cosines**

**From:**Trevor Kenchington

**Date:**2002 Oct 27, 21:11 -0400

Herbert Prinz wrote: > While removing Dan Allen's confusion about the cosine theorem, Trevor > Kenchington adds some of his own: > > >>If it were true that: >> >>cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) >> >>and: >> >>cos(c) = cos(a)*cos(b) + sin(a)*sin(b)*cos(ab) >> >>then it would necessarily be true that: >> >>sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) = cos(a)*cos(b) + >>sin(a)*sin(b)*cos(ab) >> >>since both are equal to cos(c). And so we would have to suppose that >>sin(a)=cos(a), which is obviously absurd. >> >> > > Trevor is meaning to say > > sin(a) = cos(b) Just for the record, I meant nothing of the kind and did mean exactly what I wrote. Herbert has, however, pointed out that the two equations for cos(c) could be simultaneously true if sin(a) was equal to cos(b), as well as if sin(a) was equal to cos(a). I had missed the one while noting the other. He has also pointed out that sin(a) _is_ equal to cos(b) for a limited set of values of a and b. But then again, sin(a) is also equal to cos(a), provided that a is 45 degrees. Clearly, what I meant to say (but failed to do explicitly) was that the two equations for cos(c) cannot both be universally true over all possible values of a and b. On that, I think Herbert and I are in full agreement. Trevor Kenchington -- Trevor J. Kenchington PhD Gadus{at}iStar.ca Gadus Associates, Office(902) 889-9250 R.R.#1, Musquodoboit Harbour, Fax (902) 889-9251 Nova Scotia B0J 2L0, CANADA Home (902) 889-3555 Science Serving the Fisheries http://home.istar.ca/~gadus