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    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
    
    
    

       
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