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    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
    
    
    

       
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