Nowadays, when one searches for little-known trigonometric functions, one usually finds a list containing the versine, coversine, vercosine, and covercosine. When using this list, $1+\cos(x)$ is given the name vercosine. However, I have not found any references to this before the year 2009, when it was added to Wikipedia.

In contrast, I can easily find references to the versine and coversine back to at least Cauchy's 1821 "Analyse Algébrique". Cauchy calls the coversine the cosinus versus, which formula is $1-\sin(x)$. A more detailed history can be found in Miller's Earliest Known Uses. I have seen this function also called the vercosine, for instance in MathHorizon (2006) and the Science Vocabulary builder by Johnson O'Connor in 1956.

Is there any source before 2009 where the function $1 + \cos(x)$ was given a name?

  • $\begingroup$ In which 2009 reference did you find the mention of $1+\cos(x)$ as vercosine? Also, $1-\sin(x)$ being called vercosine are a result of translation errors. $\endgroup$ – Big Brother Jul 5 at 7:34
  • 1
    $\begingroup$ Versine much predates Cauchy, see Miller's Earliest known uses:"In Practica geomitrae, Fibonacci used the term sinus versus arcus... Regiomontanus (1436-1476) used sinus versus for the versed sine in De triangulis omnimodis (On triangles of all kinds; Nuremberg, 1533)". The concept is much older and goes back to Surya Siddhanta (c. 400 AD). Extending terms people are vague on the meaning of by association is quite common. $\endgroup$ – Conifold Jul 5 at 8:57
  • $\begingroup$ The 2009 reference is Wikipedia. $\endgroup$ – user743012 Jul 5 at 15:29
  • $\begingroup$ I am aware that sinus versus and cosinus versus have a long history, which is why I wasn't that concerned about giving the full history. But that's still helpful. $\endgroup$ – user743012 Jul 5 at 15:31

The historical name of this function is the suversed sine, suversine, or susinus versus, and is abbreviated $\operatorname{suvers}(x)$. (Similarly, the function $1 + \sin(x)$ is called the cosuversine or sucoversine.) The earliest use of this name may be in 1801 by Joseph de Mendoza y Rios.

We see in Gregory and Law's 1862 Mathematics for Practical Men,

  1. The suversine of an arc is the versed sine of its supplement, as $A'D$.

In Snowball's 1837 The elements of plane trigonometry, we see

  1. The versed sine of the supplement of the $\angle BAC$ is called the suversine of the $\angle BAC$;

or, $\operatorname{suversin}\angle BAC = \operatorname{versin}(180^\circ -\angle BAC).$

Curiously, this source does not reference the coversine (or sucoversine) at all.

Thomas Kerigan (1828) uses the term "versed sine supplement".

In The Monthly Review, For October, 1806. Art. II, it is suggested that Joseph de Mendoza y Rios is the originator of the name, from his Tables for Navigation (1806):

We have mentioned certain terms, suversed, sucoversed, &c. which are novel in mathematical language; and M. Mendoza is, we believe, the author of the "callida junctura."— We subjoin the values of these lines, from which our readers may easily discern the reason for their denomination. Suppose the radius 1 \begin{align} \text{then} \operatorname{versin.} A &= 1 - \operatorname{cos.} A \\ \operatorname{suvers.} A &= 1 + \operatorname{cos.} A \\ \operatorname{covers.} A &= 1 - \operatorname{sin.} A \\ \operatorname{sucovers.} A &= 1 + \operatorname{sin.} A \end{align}

Actually, the suversed sine and sucoversed sine are mentioned in Mendoza's earlier 1801 set of tables. However, no justification for the terms is given, which suggests that they were either already in common use or had sufficiently obvious names.

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