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    Spherical Law of Cosines
    From: Dan Allen
    Date: 2002 Oct 18, 10:05 -0700

    On Friday, October 18, 2002, at 07:53 AM, David Weilacher wrote:
    > Please define little (a), (b), (c), and (ab), for me.
    With pleasure!
    cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * cos(ab)
    a is the length of the first side of the spherical triangle.
    b is the length of the second side of the spherical triangle.
    ab is the angle between the two sides a and b.
    c is the length of the side opposite the angle ab.
    Here is an example of a great circle distance between
    San Francisco (SF) and Salt Lake City (SLC).  All angles
    and lengths are expressed in degrees for this example,
    and North and West are positive.
    lat1 = 37 degrees
    lon1 = 122 degrees
    lat2 = 40 degrees
    lon2 = 112 degrees
    a is the co-latitude of lat1, or 90-lat1 or 53 degrees.
    b is the co-latitude of lat2, or 90-lat2 or 50 degrees.
    ab is the difference of the longitudes, or lon1-lon2 or 10 degrees.
    Solving for c one learns the distance in degrees between SF and SLC,
    which is about 8.375 degrees.  To get a distance we understand,
    multiply degrees times 60 nmi per deg and you get 502.5 nmi.
    One then can re-substitute back in and use the formula again to
    determine the initial great circle course.  In this case we
    use a as co-latitude of lat1, b is the distance we just
    computed (8.375 degrees -- make sure to use degrees), c is
    the co-latitude of lat2, and the angle ab is the initial
    course.  We have rotated the whole spherical triangle around
    to put the sides a,b, and c where we know their values,
    leaving ab to be solved for.  Substituting and solving one
    determines that the initial course is 65.9588 degrees.
    In sight reduction:
    a is the co-latitude of the assumed position
    b is the declination of the body (say the sun)
    ab is the hour angle of the body
    c is the altitude of the body
    Note too that there are equivalent ways of writing the spherical
    law of cosines.  One can write it as
    cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
    where the pairs of sin/cos of a/b are switched.  This allows one to do
    great circle calculations using latitudes directly,
    without using co-latitudes.  The origin is moved from the pole to
    the equator, so to speak.  This form is often handier but the first
    version is the canonical version.
    Hope this helps.

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