# Definition of the $\sec$ function

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?
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mathworld.wolfram.com/Secant.html has some information – Alex Apr 8 '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 8 '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 8 '14 at 13:54
@StephenMontgomery-Smith \$\csc\$ – Git Gud Apr 8 '14 at 13:55
The Russian Wikipedia article on trigonometric functions mentions it and uses the same abbreviation. So does the German one. – MJD Apr 8 '14 at 13:56

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.
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 8 '14 at 13:56
Slovenian Wikipedia mentions it. The article says that in Slovenian it is Sekans and is abbreviated $\sec$. – MJD Apr 8 '14 at 14:00
@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