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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