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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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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)
>

gives

(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
supple).

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
> 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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