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I was preparing for a calculus exam and I came across the Wikipedia article for all the trigonometric identities. There I came across some terms that I had never seen before. They were: $\text{versin}(x)=1-\cos(x)$, $\text{coversin}(x)=1-\sin(x)$, $\text{vercosin}(x)=1+\cos(x)$ and other similar ones.

In the article it states that these were historically used but nowadays they have no real use. Why is that? Why are these not used anymore?

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    $\begingroup$ Could that be a question for hsm.stackexchange.com? $\endgroup$
    – Martin R
    Commented Jan 24, 2021 at 15:45
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    $\begingroup$ I woul assume it is much simpler to write $1-\cos(x)$ than $\operatorname{versin}(x)$. $\endgroup$
    – user
    Commented Jan 24, 2021 at 15:45
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    $\begingroup$ It's not that they're not "used", but that nobody cares to give them special names. For example, $\cos^2(x) = (1+\cos(2x))/2$ and $\sin^2(x) = (1-\cos(2x))/2$, so you could write the numerators in these formulas in terms of vercosin and versin, but it's irrelevant. Just write $1 \pm \cos(2x)$ and that's fine. It certainly is clearer. Anyway, the more important page about wacky trigonometric functions is theonion.com/…. $\endgroup$
    – KCd
    Commented Jan 24, 2021 at 15:46
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    $\begingroup$ Perhaps special tables were initially created for these functions because 100+ years ago there were no calculators, and therefore a need was felt to have a look-up tables of values of $1 \pm \cos(\theta)$ and similar stuff. But with calculators widely available to students for 40+ years, such data is unnecessary to collect in a separate way. $\endgroup$
    – KCd
    Commented Jan 24, 2021 at 15:48
  • $\begingroup$ @KCd Hmmm. That is quite possible. In the article it mentions that these identities were used for sailing and related measurements. I think your second comment touches up on the really nicely. $\endgroup$
    – Ski Mask
    Commented Jan 24, 2021 at 15:52

2 Answers 2

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As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.

These days we walk around with devices capable of approximating the versine to a degree of accuracy that would in the past be considered an extraordinary degree of accuracy and capable of looking up tables the size of books for exact results.

That is the practical aspect, the other aspect is that it has fallen out of taste, and there is no accounting for taste so take it as you will.

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The versine function is well documented in this Wikipedia article: https://en.wikipedia.org/wiki/Versine

The versine or versed sine is a trigonometric function found in some of the earliest (Vedic Aryabhatia I) trigonometric tables. The versine of an angle is 1 minus its cosine.

It is very likely that people need all sorts of trig functions to write trigonometric tables in the old days. But now we have computers.

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