Maybe it makes sense if we think about invertible functions as a group with
composition as the binary operation, but try telling that to a trig student.
:-) Personally, I tell my students that this notation exists and then give
it a wide berth in class, sticking to asin, acos, etc., to be consistent
with the notation in the technology we use (mostly Maple). Admittedly it's a
mess, but given the length of time it takes historically for notations to
become standardized, it's probably not going to go away in our professional
lifetimes.
Chuck Lindsey, Ph.D. clindsey@fgcu.edu
Director of General Education
Associate Professor and Program Leader, Mathematics
Florida Gulf Coast University
10501 FGCU Blvd South
Fort Myers, FL 33965-6565
Phone: (941) 590-7168 FAX: (941) 590-7200
http://itech.fgcu.edu/faculty/clindsey
> -----Original Message-----
> From: mahlerp@middlesex.cc.ma.us [SMTP:mahlerp@middlesex.cc.ma.us]
> Sent: Friday, March 19, 1999 4:52 PM
> To: DTISCHER@metropo.mccneb.edu
> Cc: mathedcc@archives.math.utk.edu
> Subject: Re: [MATHEDCC] Technology and Trig
>
> DTISCHER@metropo.mccneb.edu writes:
> >I am working on using technology to address the misconceptions that I
> >fight
> >most frequently with students in a trigonometry course. I am hoping that
> >there are materials somewhere that have already dealt with the issues.
> If
> >anyone knows of software, web sites, etc. that might be possibilities,
> >please let me know. The issues I am focusing on deal with errors in
> >interpreting notation and are:
>
> Couldn't resist an opinion on this one.
> >
> > 1. sin(superscript -1)x does not mean csc x even though
> >sin(superscript 2)x means sin squared of x.
>
> This is hard for students because the notation is terrible. Statement 1
> illustrates the inconsistent use of exponents here. What would (sin
> x)^(-1) mean? and since (sin x)^2 == sin^2 x, why isn't this true if the 2
> is replaced by -1?
>
> In fact, what would sin(superscript -2)x mean?
>
> Given these inconsistencies, sin(superscript -1)x is not a very good
> notation. I like the suggestion of using asin, or arcsin, or invsin,
> instead of the -1 exponent. Of course I realize textbooks are permeated
> with the more sophisticated notation.
>
> I realize that x^(-1), x a number, is nice in that this represents both
> division and the inverse operation for multiplication. But the temptation
> to label the inverse of any function f as f^(-1) is perhaps not a good
> idea. The symbol x^y, x, y real has a consistent interpretation for all x
> and y, with rules for manipulating expressions of this form, but if f is a
> function, then f^y only has limited interpretations, and they are not
> consistent. ... Please don't think I'm advocating a different notation for
> the inverse of a function - it's too late. And f^(-1)(x) is a handy
> notation for mathematicians, but it is usually used in a context of
> discussing functions and not real numbers. Our poor students see both
> notations at the same time, guaranteed to confuse.
>
> Thus I find it hard to blame students, who are used to the power of
> algebra being in its notation.
>
> All of the suggestions made on the list would help. I also tell my
> students that sin^(-1)x is a very special case, and if it meant csc x, we
> wouldn't use both notations.
> >
> > 2. The statement: sin(superscript -1)x = theta should be translated
> >to sin theta = x.
>
> I'm not sure about the point here. This is an almost-true statement.
> Adding the suitable restrictions on theta would make it true. And taking
> "should" in the sense of "is very helpful to" I do use rewriting, and
> stress the restrictions as being in Quadrant I and IV for sine and
> tangent, and I and II for cosine.
>
> And I prefer to use equivalencies for the other 3 inverses. arcsecant(x)
> = arccos(1/x) for example. It really isn't worth spending a lot of time on
> these last 3 functions. (inviting contrary opinions, of course)
> >
> >and 3. When simplifying product to sum answers, realize that 1/2(sin 4x
> >+ sin 2x) does NOT equal sin 2x + sin x. (I have just completed
> >graphing y
> >= 1/2(sinx) and y = sin 2x and they do that successfully but don't
> relate
> >it to this!)
>
> Computer languages were already mentioned on this thread. So it might be
> helpful to compare 1/2 sin 4x to 1/2 sqrt(2x), or 1/2 log 2x (of course
> logs usually follow trig). Of course most students have troubles with 1/2
> sqrt(2x)! (That wasn't factorial. :-) )
>
> I think one of the hardest things for student is what can NOT be done.
> (2x + 3y)/2 = x + 3y is a freshman mistake, but it doesn't go away for my
> developmental students. (Naturally I explain through counter-examples why
> this doesn't work, and the situations in which factoring a common factor
> would let it work.)
>
> I don't have an easy answer for this one. I do say that in general a
> coefficient cannot affect the argument of a function ... until we get to
> logs, and then only in special, non-intuitive ways.
>
>
> My closing opinion is to not belabor any of these points. I'd rather see a
> student be a little fuzzy on these and be able to manipulate vectors, say,
> than spend a lot of time on points like this and have a student leave a
> trig or precalc course and not know the power of trig to solve problems.
>
> Philip Mahler
> Middlesex Community College
> Bedford, MA
>
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