Re: Perhaps this is...
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> > but how come the graph of sin^-1(x) is different from the graph of
> > 1/sin(x)? If x=1 and I try to calculate these two functions, they yield
> > different values.
> >
> > Aren't they the same function? They seem equivalent.
> >
> > Mark R.
> > Hunter HS, 11th Grade, NYC
> > TI-83
> >
> Your first relationship is the inverse-sine of x. The second is the
> inverse of the sine of x. These are not the same. Think of the second,
> inverse of sine of x as when you take off your sock and shoe, reverse
> your
> sock inside out and put the shoe back on over it. Think of the first
> case, inverse-sine of x, as when you take off your sock and shoe replace
> the shoe and then put your sock over your shoe. Both cases your reversed
> your sock but the outcomes are not the same.
>
> Congratulations on using your calculator as a place to start thinking
> from. Too many of us use calc as a place where thinking ends.
>
> stay connected,
> Elwood
The real problem here is inadequate notation -- a problem that often
comes up in mathematics because of the patchwork way in which
mathematical notation has evolved over the centuries. Division is the
inverse of multiplication -- dividing by a number x undoes the effect of
multiplying by x. If we use exponents, multiplying by x^-1 undoes the
effect of multiplying by x^1.
The problem now becomes, how do we undo the effect of a more
complicated function? The answer is that it is not always possible, but
that's another story! It _is_ possible to find a function that undoes the
effect of sine (if we confine ourselves to a limited set of inputs), and this
function is called the inverse sine function. BUT, what do we _name_ this
function? It has been traditional to name this inverse function sine^-1,
even though we don't "multiply by sine" when we evaluate the sine of a
function (!). The old-fashioned notation arcsin (for "the arc length needed
to produce a given value of sine") was more honest, but it didn't transfer
well to other function-inverse situations. I am afraid that we are stuck with
f^-1 for the foreseeable future.
But think clearly about this: if we start with a given number, take
its
sine, and then take the reciprocal of the result, we probably aren't going to
get our original number back! (That's why 1/sin(x) is not the inverse of the
sine function). The real inverse of the sine function (conventionally called
sin^-1) is built into most scientific calculators and labeled with that name.
The fact that it is so easy to play with functions and their inverses with
calculators such as the TI-8x's is the reason why it is possible to teach
more math, more quickly these days! A whole lot more thinking going on...
RWW Taylor
>The plural of mongoose
<
National Technical Institute for the Deaf > begins with p.
<
Rochester NY 14620
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