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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: Cosine Method Revisited
From: Chuck Taylor
Date: 1996 Dec 15, 20:39 EST

```Gordon Talge, Oliver Jacobson, and perhaps others have commented
that there is no ambiguity in the cosine formulas for sight
reduction. At the risk of belaboring the point, I'd like to be a
little more precise.
Referring to the following form of the equations:
sin Hc = (cos LHA * cos Lat * cos Dec) + (sin Lat * sin Dec)
cos Z  = (sin Dec - (sin Lat * sin Hc)) / (cos Lat * cos Hc)
Note first that I show LHA instead of t. There are two possible
sign conventions to use for N/S Lat & Dec. If you choose North
Lat and North Dec as + and South Lat and South Dec as -, the
result is Zc, which can be converted to Zn = Azimuth by the
following formulas:
LHA > 180, Zn = Zc
LHA < 180, Zn = 360 - Zc
If, on the other hand, you choose a different sign convention,
namely that Lat is always positive and that Dec is positive if it
has the same name as Lat (both N or both S), and negative
otherwise, then the result is Azimuth Angle Z, which is converted
to Azimuth Zn by the following formulas:
Lat N & LHA >180, Zn = Z
Lat S & LHA >180, Zn = 180 - Z
Lat S & LHA <180, Zn = 180 + Z
Lat N & LHA <180, Zn = 360 - Z
Using these conventions, your calculator will give you an
unambiguous result. Why? It's largely a matter of happy
circumstance. The signs of the various trig functions just happen
to work out for the best. Check it out for yourself! Or perhaps
someone else has a better explanation why it works....
Chuck Taylor
Everett, WA
ctaylor@XXX.XXX
```
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