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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: Gordon Talge
Date: 1996 Dec 15, 15:53 EST

```In replay to cosine method. I was not implying that there was an ambiguity
problem with it, what I ment was that if you are not careful with trig, stuff
like that can come up.
The cosine method is not the only trig formula that will work, although it
is the fundental basic one. There are many others, check Bowditch, older
editions.
The problem with the cosine formula with hand calculations is the logs of
negative numbers.
To use sin h = sin L sin d + cos L cos d cos t with a calculator is straight
forward, but with trig tables it is alot of work.
Log trig make it easer, but you have the problem that ( sin L sin d ) could be
negative. To solve this they used the haversine formula :
hav z = hav (L ~ d) + cos L cos d hav t. Since the haversine is always positive
and so is the cosine, there is not any problem with the logs.
Here z is zenith distance.
The fundamental formula for Z is sin Z = sin t cos d sec h. Now here we have
a problem with the ambiguity. This formula is fundamental also. Now there are
better trig formulas for Z, such as outlined in the nautical almanac.
As far the difference between a circle of position and line of position.
The circle of position looks like a line of position locally. You can
get away with representing circles as straight line on charts for small
distances in most area of the world. This is the whole idea behind mecator
charts.
Godon
,,,
(. .)
+-----------------------ooO-(_)-Ooo----------------------+
|  Gordon Talge WB6YKK            e-mail: gtalge@XXX.XXX  |
|  Department of Mathematics      QTH: Loma Linda, CA    |
|  Mt. San Jacinto College        Lat.  N  34? 03.1'     |
|  San Jacinto, CA                Long. W 117? 15.2'     |
+--------------------------------------------------------+
```
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