Definition of the $\sec$ function

Question

I am a postgraduate student of mathematics from Slovenia (central Europe) with quite some experience in mathematics. While answering questions on this site, I often encounter the function $\sec(x)$ which is, as I understand, defined as $\sec(x) = \frac1{\cos x}$. During my studies, I never encountered this function.

I am wondering two things:

How widespread is the notation $\sec x$? As I see it, it is completely standard notation in US schools, but not as common in Europe. Is there a historic reason behind this dichotomy?
Is there a reason for calling the function $\sec$? Is there some geometric interpretation behind the name?

http://mathworld.wolfram.com/Secant.html has some information — Alex, Apr 08 '14 at 13:47
In Britain (where I grew up) and the US (where I live), the use of the notation $\sec$ is very widespread. There is also the $\cosec$ or cosecant, which is the reciprocal of the sine function. The only pattern I could see is that for trig functions the prefix "co-" indicates that the function is decreasing on the range $[0,\frac\pi2]$. — Stephen Montgomery-Smith, Apr 08 '14 at 13:51
@StephenMontgomery-Smith The pictures that I linked to below explain the use of tangent and sectant coming from lines tangent to and lines that cut the circle. The second one also justifies the co-prefix. — Fly by Night, Apr 08 '14 at 13:54
The Russian Wikipedia article on trigonometric functions mentions it and uses the same abbreviation. So does the German one. — MJD, Apr 08 '14 at 13:56
@MJD In this case, I don't find wikipedia to be too trustworthy. The slovenian Wikipedia also mentions the $\sec$ function, but I have enough experience to confidentelly say that this notation is not used in Slovenia. I suspect that the appearance in the wikipedia is already influenced by the english wikipedia. — 5xum, Apr 08 '14 at 14:00
Can the voter explain the downvote? What is wrong with my question? — 5xum, Sep 23 '14 at 06:57

score 8 · Accepted Answer · answered Apr 08 '14 at 13:50

8

First of all, the use of $\sec$ is standard in the United Kingdom which is part of Europe. The fact might be that $\sec$ is used in English-speaking countries, of which Slovenia is not.

The word Secant comes from a Latin word meaning "to cut". This picture shows the definition of sine, tangent and secant. It justifies the naming of both the tangent and secant functions. This picture shows the co-functions, e.g. co-sine, co-tangent and co-secant.

answered Apr 08 '14 at 13:50

Fly by Night

32,272

Thank you. So it is true that the notation is, as I said, "not as common" in Europe (only really common in the UK), although a better description would therefore be "not as common in non-English speaking nations". Also, thanks for the picture, very informative! – 5xum Apr 08 '14 at 13:55
3

It is common in Portugal. I was really surprised reading the first paragraph of the question that a graduate student didn't know about $\sec$. It never crossed my mind that it isn't used world wide. – Git Gud Apr 08 '14 at 13:56
Slovenian Wikipedia mentions it. The article says that in Slovenian it is Sekans and is abbreviated $\sec$. – MJD Apr 08 '14 at 14:00
@5xum Having looked at different Wikipedia versions of the Trigonometric Functions article, $\sec$ is used in French, German, Spanish, Russian and even Slovakian (Slovenčina). – Fly by Night Apr 08 '14 at 14:00
Yes, but as I already mentioned in comments to my question: The slovenian Wikipedia also mentions the $\sec$ function, but I have enough experience to confidentelly say that this notation is not used in Slovenia. I suspect that the appearance in the wikipedia is already influenced by the english wikipedia. I find it possible that this also happens in other languages. – 5xum Apr 08 '14 at 14:02
@GitGud All of the different versions of Wikipedia I have looked at use $\sec$, and not just those with the Roman alphabet. Even Russian and Slovakian use $\sec$. – Fly by Night Apr 08 '14 at 14:02
@GitGud My field is not analysis, so my knowledge of analysis is basically a firm graduate understanding, nothing more. Still, I would consider myself quite knowledgable in the area. However, I have never encountered the $\sec$ function in any text written in Slovenian. I also find the function rather unnecesary and confuzing, but that's just my oppinion. – 5xum Apr 08 '14 at 14:11
@5xum Perhaps. Although the Slovakian Wikipedia uses $\operatorname{tg}x$ instead of $\tan x$, so I would expect it to use its own version of $\sec x$ if there was one. In my experience, I only ever studied trigonometry at school and never at University. Perhaps $\sec$, $\csc$ and $\cot$ are not taught in Slovakian schools and so you never met them. Once you get to University, you would never have needed to use them. That might explain it. – Fly by Night Apr 08 '14 at 14:14
@FlybyNight Just a FYI, I am Slovenian, not Slovakian. So no, I never met things that happen in a country I never visited. – 5xum Apr 08 '14 at 14:24
@5xum LOL! Sorry for that. No offense intended! – Fly by Night Apr 08 '14 at 14:29
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@FlybyNight None taken. It happens way to often to offend me or anyone in either of the two countries. We pay the british back by calling everyone in the UK "English", all Canadians "Americans" and half the Belgians "French". – 5xum Apr 08 '14 at 14:42
2

@5xum (And Fly by Night) My first encounter with $\sec$ and $\operatorname{cosec}$ was on page 44 of Ahlfors' Complex Analysis. I had to look it up in the encyclopaedia. I can't authoritatively speak for all of Germany, but in the parts where I went to school and university, these functions were not used (anymore, they had been used in the past). – Daniel Fischer Apr 10 '14 at 23:21

Definition of the $\sec$ function

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