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Re: Haversine- how to derive it?
From: Frank Reed
Date: 2015 May 16, 16:00 -0700

The use of the "haversine" name outside of historical discussions is mere jargon. There were a few decades in the early twentieth century when haversine tables were commonly used in navigation, but they are not important (they were also implicitly used, though not generally named as such, in many nineteenth century navigation solutions). The haversine of any angle x can be easily calculated:
hav(x)= [1 - cos (x)] / 2.
Suppose you have been given an equation that is solved with a haversine result. In other words, hav(x)=something. You might think you need "hav" tables to look up the solution or maybe an "inverse hav" key on your calculator. You might imagine that you would work out the value of that "something" and then use the inverse haversine function to get x. But it's not necessary. This equation, hav(x)=something, is exactly equivalent to
cos(x) = 1 - 2·something.
And obviously you can solve that with the usual cos tables or an "inverse cos" on a calculator. Historically, multiplying by 2 and subtracting from 1 would have been considered a lot of work in a problem that had to be worked up over and over again. But those days are gone. Today the haversine is the trig function for the "navigator who has everything". If you are a "one of each" collector, then you'll want it for your stamp book. Otherwise, it has no value. You don't need, but it's certainly nice to know the jargon. And in the event the GPS ever goes down, you'll be able to jump up and say, "Haversines Ho!!".

Incidentally, the name of the thing has a simple origin. The "versine" was historically defined as 1 minus the cosine, from "versus" for flipped, or more mnemonically, but still a little misleading, versine is the "reversed sine". But we want that quantity divided by two or "half versine" hence "haversine".

Frank Reed

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