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    Re: Tables vs. Calculators
    From: Richard Irvine
    Date: 2002 Sep 23, 11:03 +0200

    Since you mention both series expansions and the spherical cosine rule I
    thought I would just point out how this formula contains (at least) three
    for the price of one.
    If the sides of the spherical triangle are small then we can subsititute the
    first terms from the series expansion of sine and cosine
    For small x, sin x ~= x,   cos x ~= 1 - x**2/2 (1 minus x squared on 2)
    Substituting into the spherical cosine formula
    >    cos(a)*cos(b) + sin(a)*sin(b)*cos(ab) = cos(c)
    (1-a**2/2)(1-b**2/2) + abcosC = (1-c**2/2)   (I have written C instead of
    your ab for the angle opposite the side c).
    or, rearranging,
    c**2 = a**2 + b**2 -2abcosC    neglecting the term in a**2.b**2 (second
    order of small quantities)
    If we keep the sides of the triangle constant but inflate the sphere until
    its radius is so great that the spherical triangle is planar, then the
    approximation becomes exact.
    In other words this is a derivation of the planar cosine rule. This is how I
    check that I have remembered the spherical cosine rule correctly (having had
    the planar cosine rule hammered home at an age when my memory was more
    The special case that C = 90 degrees then gives Pythagoras theorem.
    > -----Original Message-----
    > From: Dan Allen [mailto:danallen46{at}ATTBI.COM]
    > Sent: Friday, September 20, 2002 11:52 PM
    > To: NAVIGATION-L{at}listserv.webkahuna.com
    > Subject: Re: Tables vs. Calculators
    > On Friday, September 20, 2002, at 11:32 AM, Chuck Taylor wrote:
    > > I could even reproduce the sines and cosines if I wanted to trouble
    > > myself with going through a Taylor series expansion.
    > You're lucky: you've got a series expansion named after yourself,
    > but the rest of us (except for Maclaurins) are just out of luck... ;-)
    > Seriously, I totally agree with Chuck.  I can and do have the basic
    > formula for great circle nav and sight reduction memorized because it
    > is so simple:
    >    cos(a)*cos(b) + sin(a)*sin(b)*cos(ab) = cos(c)
    > Many of our nav problems boil down to using this simple formula.
    > For example, for great circle problems with arguments in degrees
    >    IF a = 90 - lat1
    >     & b = 90 - Lat2
    >     & ab = Lon2 - Lon1
    >    THEN
    >       c = GC Distance in degrees, or multiply it by 60 for
    > nautical miles
    > or for sight reduction with arguments in degrees
    >    IF a = 90 - estimated latitude
    >     & b = 90 - declination
    >     & ab = LHA = GHA - estimated longitude
    >    THEN
    >       90 - c = altitude
    > This formula is easy to program into calculators, or easy to write
    > down on a piece of paper and do by hand with a basic scientific
    > calculator.  This is the essence of self-reliant navigation.
    > Dan

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