Trigonometry Terminology

Date: 10/24/2001 at 22:09:26
From: Scott VanHorn
Subject: Arc sin

Dear Dr. Math,

Recently in my Pre-Calculus math class, we have been dealing with 
functions such as sine, cosine, and tangent. We are currently 
confused with some math terminology, and you may be able to help us.

Our question is: Why is the term "arc" used for the inverse of sine, 
cosine, and tangent instead of just saying the function to the -1 
power?

Any help that you can provide us is greatly appreciated.

Sincerely,
Scott VanHorn

Date: 10/24/2001 at 23:38:55
From: Doctor Peterson
Subject: Re: Arc sin

Hi, Scott.

Trigonometry terminology is somewhat archaic in general, and doesn't 
fit well with algebraic notation. There are some good reasons for both 
starting out different, and being kept different, though it's not an 
ideal situation.

As for its origin, "arcsin" simply means "the arc whose sine is ...". 
This is pretty straightforward, and was especially so before function 
notation and its "f^-1" was invented. It appears that you have been 
taught only the "arcsin" form, but in fact "sin^-1" is quite common.

As for its continuation, the use of sin^-1 collides with the tradition 
of writing sin^2(x) for the square of the sine, another holdover from 
early usage. One of the two has to give way, unless we keep both, and 
trust the reader to see whether sin^-1(x) means the inverse sine or 
the inverse of the sine, the cosecant. I personally prefer arcsin 
(especially in writing for Dr. Math) for this reason. This notation 
also retains its popularity (and might even be growing) because in 
computer programs it is easier to name a function "atan" rather than 
something clumsy like "tan_inverse." In my mind, it makes good sense 
to give the inverse functions names of their own, like arcsin, rather 
than always treating them as mere inverse functions.

If you study the history of the notation, as in Jeff Miller's 

  Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions
  http://jeff560.tripod.com/trigonometry.html   

you find that early forms of "arcsin" arose in the 1700s, and the 
sin^2 notation came at the same time; sin^-1 was introduced in 1813. 
I presume that the inverse function notation in general was an 
immediate precursor to that (or else arose from it), but I can't 
locate that information. The above page says that arcsin is usual on 
the continent, while sin^-1 is used in England and America. I don't 
know whether that is still true.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/