As recently as 1984, the amateur astronomy magazine Sky & Telescope was singing the praises of the haversine formula, which is not only useful for terrestrial navigation but also for celestial calculations. (On the other hand, the haversine formula does not do a very good job with angles that are close to 90 degrees, but the spherical law of cosines handles those well.) The haversine formula could yield accurate results without requiring the computationally expensive operations of squares and square roots. The haversine formula is a re-formulation of the spherical law of cosines, but the formulation in terms of haversines is more useful for small angles and distances. The haversine formula is a very accurate way of computing distances between two points on the surface of a sphere using the latitude and longitude of the two points. But the haversine may have been more important in more recent history, when it was used in navigation. The versine is a fairly obvious trig function to define and seems to have been used as far back as 400 CE in India. If you have a computation involving the square of sine or cosine, you can use a haversine or havercosine table and not have to square or take square roots. memorization of one of the endless list of trig formulas you learned in high school) shows that 1-cos(θ)=2sin 2(θ/2). A little bit of trigonometric wizardry (a.k.a. (It is not defined for 0 either, but that is an easy case to deal with.) Another advantage to the versine and haversine is that they can keep you from having to square something. Versine ranges between 0 and 2, so if you are using log tables to multiply with a versine, you don't have to worry about the fact that the logarithm is not defined for negative numbers. The bonus trig functions also have the advantage that they are never negative. In many cases, this wouldn't matter, but it could be a problem if the errors built up over the course of a computation. And a table with only three significant figures of precision would not be able to distinguish between 0 degree and 1 degree angles. If you had three significant figures in your cosine table, you would only get 1 significant figure of precision in your answer, due to the leading zeroes in the difference. To illustrate, the cosine of 5 degrees is 0.996194698, and the cosine of 1 degree is 0.999847695. If you were doing a computation that had 1-cos(θ) in it, your computation might be ruined if your cosine table didn't have enough significant figures. Near the angle θ=0, cos(θ) is very close to 1. Versine and haversine were used the most often. The secret trig functions, like logarithms, made computations easier. When each operation takes a nontrivial amount of time (and is prone to a nontrivial amount of error), a procedure that lets you convert multiplication into addition is a real time-saver, and it can help increase accuracy. It sounds cumbersome now, but doing multiplication by hand requires a lot more operations than addition does. Then you'd use your log table to find out which number had that logarithm, and that was your answer. If you wanted to multiply two numbers together using a log table, you would look up the logarithm of both numbers and then add the logarithms together. In other words, logarithms make multiplication into addition. One handy fact about logarithms is that log b(c×d)=log bc+log bd. Numberphile recently posted a video about Log Tables, which explains how people used logarithms to multiply big numbers in the dark pre-calculator days. But these seemingly superfluous functions filled needs in a pre-calculator world. Why did they even get names?! From a time and place where I can sit on my couch and find the sine of any angle correct to 100 decimal places nearly instantaneously using an online calculator, the versine is unnecessary. They're all just simple combinations of dear old sine and cosine. I must admit I was a bit disappointed when I looked these up. Hacovercosine: hacovercosin(θ)=covercosin(θ)/2 Whether you want to torture students with them or drop them into conversation to make yourself sound erudite and/or insufferable, here are the definitions of all the "lost trig functions" I found in my exhaustive research of original historical texts Wikipedia told me about. Not pictured: vercosine, covercosine, and haver-anything. Excosecant and coversine are also in the image. The versine is in green next to the cosine, and the exsecant is in pink to the right of the versine. (It's well known that you can shake a stick at a maximum of 8 trig functions.) The familiar sine, cosine, and tangent are in red, blue, and, well, tan, respectively. A diagram with a unit circle and more trig functions than you can shake a stick at.
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