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I was browsing through a Wikipedia article about the trigonometric identities, when I came across something that caught my attention, namely forgotten trigonometric functions.

The versine (arguably the most basic of the functions), coversine, haversine and exsecant formulas had once been utilised for navigational purposes, prior to GPS tracking systems. However, recently, they have become less common in modern mathematics and beyond. Why is that?

Here is a link to a PDF file describing all of these now-obsolete trig functions:

Rodrigo de Azevedo
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LÜHECCHEgon
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Those functions are much less used than before for one reason: the advent of electronic computers.

Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.

For instance, to compute $\log\sqrt{a^2+b^2}$ when $\log a$ and $\log b$ are known, you could find $\theta$ such that $\log\tan\theta=\log\frac ba=\log b-\log a$, then $\log\sqrt{a^2+b^2}=\log a+\log\sqrt{1+\tan^2\theta}$ and $\log\sqrt{1+\tan^2\theta}=\log\frac{1}{\cos \theta}=-\log\cos\theta$. There are many similar formulas.

For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $\sin^2(\theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.

Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.

All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.

Jean-Claude Arbaut
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    The log tables we used at school were four figure. I repeatedly got marked down (indirectly) for using five figure tables: I gave my answers to four figure precision, rather than the three I should have done if I were using four figure tables – JCRM Apr 17 '19 at 12:28
  • it included trig and otehr tables to -- 48 pages apparently https://www.amazon.co.uk/Four-Figure-Tables-C-Godfrey/dp/0521050979 – JCRM Apr 17 '19 at 13:21
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    Oddly enough, I bought a log book 5 years back. I have no idea what my 11 year old self was thinking, but it was pretty cool. – For the love of maths Apr 17 '19 at 13:49
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    Just re-opened the log table. 6 digits of accuracy for logs, anti-logs, trig functions, log of trig functions and so much more at 47 cents in the US. 61 pages including "some" formula and useful data. – For the love of maths Apr 17 '19 at 14:06
  • Whilst this sounds reasonable, no evidence is given. Mostly in maths we never evaluate expressions, so existence of tables, calculators or computers is irrelevant. – Keith Apr 17 '19 at 21:37
  • Note that things like the haversine formula are still useful in a computer program; they just take an extra subtraction and multiplication by $\frac12$ since we don't have a direct haversine function. – David K Apr 18 '19 at 00:47
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    @Keith No evidence, but I have a hunch that the additional functions, as well as the tables, were never meant for higher mathematics, but were intended for practical applications in which one does have to evaluate expressions. – David K Apr 18 '19 at 00:51
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    Completely unrelated, but physical log tables have actually lead to at least one important discovery in mathematics: Benford's Law. As someone who used to work as an archaeologist, I sometimes morn the loss of our physical implements. :\ – Xander Henderson Apr 18 '19 at 03:32
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    @MohammadZuhairKhan: When you want to challenge the "no calculators" rule ... – Joshua Apr 18 '19 at 03:47
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    @JCRM I had the same problem. The recommended / standard log tables were 4 figure but my father gave me his 5 figure tables. I also got criticized for answers that were too good. I had the same issue with a non-standard slide rule. My parents did not want to buy another set of tables or slide rule when I already had better ones. – badjohn Apr 18 '19 at 14:00
  • MohammedZuhairKhan… I have a paper for you related to log tables. I have improved the standard Irish four place table. Please contact me directly, richard1941 (at) gmail.com. – richard1941 Apr 24 '19 at 03:14
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Historically, those oddball functions were used primarily in navigation to reduce sextant readings and times to latitude and longitude. With the advent of radio, the need for that was greatly reduced, as we have radio direction finding, loran, and, more recently, GPS.

richard1941
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  • There was quite a long gap (40 odd years?) between the advent of radio communications and radio direction finding, etc. – Rob Arthan Jul 29 '22 at 21:09
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It’s true that those functions are rarely used today for the reasons set out above. But, they are still used. For example, there are many yachts cruising around the world and while most, if not all, use gps, I don’t think too many would not have backup methods of navigation, meaning sextant, watch, almanac and tables to evaluate the celestial sight. I use versines, log vers, log cos, log sin tables. The computation is straightforward and that is the whole point of the functions. They make what would be otherwise very challenging calculations easy to do on paper.

David
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