# 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 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 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 at 13:54
@StephenMontgomery-Smith \$\csc\$ –  Git Gud Apr 8 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 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.

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.

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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 8 at 13:55
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 at 13:56
Slovenian Wikipedia mentions it. The article says that in Slovenian it is Sekans and is abbreviated $\sec$. –  MJD Apr 8 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 8 at 14:00
@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 8 at 14:42