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/
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