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Natural Cotangent Rule

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Use it when the natural cotangents of the two bearings differ by 1. (cot X = 1/tan X). You can either find the cotangent of each relative bearing and compare them, or you could look in Bowditch Table 505 for a list of natural cotangents.
 
What it tells you is that the distance run is equal to the distance from the object when it is abeam.
 
This works because... Bowditch says so. The mathematics involved are a little complex, but here's a shot at it, the explanation to follow may be entirely wrong, but this sounds like it should make sense. The example and diagrams should be perfectly correct though.
 
tan x = opposite / adjacent
cot x = adjacent / opposite
(see right for further explanation)

natural_cotangent.jpg