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Re: cosine/haversine method
From: Carl Herzog
Date: 2001 Feb 09, 9:03 AM

```Russell:

A variation on versine/cosine method is used by Reed's in our Nautical
Ephemeris. While sight reduction by this method requires a little more math
than H.O. 229, the advantage is that no assumed position is required and all
the necessary tables fill fewer than 20 pages.

We use ABC tables for determing azimuth. You can determine it with a calculator
or a set of trig tables, using the formula:

Sin Az = (Sin LHA x Cos Dec)
Cos HC

ABC tables however are only about 5 pages long and easier to use with just
pencil and paper. Your entering arguments are LHA and Lat in A; LHA and Dec in
B. C converts A +/- B into Zn. ABC tables are very easy to use.

As for HC, the versine formula is:

Vers Zenith Distance = Vers LHA x Cos Lat x Cos Dec + Vers (Lat � Dec)

(Note that using versines, if Lat and Dec have the same name, they should be
subtracted, and if they have opposite names, they should be added.)

To elminate the need for multiplication, Reed's tables provide log versines and
log cosines alongside the natural trig functions.  Hence, the formula becomes:

Natural Versine Zenith Distance = Nat Vers(Log Vers LHA + Log Cos Lat + Log Cos
Dec) + Nat Vers (Lat � Dec)

As I said this does require extra steps, but the biggest advantage is that all
the requisite tables fit into a book small enough to be kept in your sextant
box.

--
Carl Herzog
Editor, Reed's Nautical Almanacs

www.reedsalmanac.com
carlzog@reedsalmanac.com
-------------------------------------------
120 Fulton Street
Boston, MA 02109
Ph# (617) 227-1300
Fax# (617) 268-5905
```
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