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    Re: Spherical Law of Cosines
    From: Trevor Kenchington
    Date: 2002 Oct 26, 11:10 -0300

    Dan Allen wrote:
    > 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)
    If it were true that:
    cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
    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) +
    since both are equal to cos(c). And so we would have to suppose that
    sin(a)=cos(a), which is obviously absurd.
    The correct equation of this pair is Dan's "cononical version" (unless I
    am wildly off base). His alternate should, I suspect, be written:
    cos(c) = sin(A)*sin(B) + cos(A)*cos(B)*cos(ab)
    where A=90-a and B=90-b. Stretching memory back to high-school
    triginometry, I think it is true that sin(90-a)=cos(a), making this form
    ofthe alternate identical to the canonical version. Of course, provided
    one is careful over using (e.g.) latitudes rather than co-latitudes with
    the alternate, you could forget about explicitly subtracting anything
    from 90. But remembering the alternate version as a solution to the
    spherical triangle could get you into serious confusion.
    Trevor Kenchington
    Trevor J. Kenchington PhD                         Gadus@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

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