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### Trigonometry Terminology

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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/
```
Associated Topics:
High School Trigonometry

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