Path: utzoo!attcan!uunet!cs.utexas.edu!news-server.csri.toronto.edu!utgpu!watserv1!ria!uwovax!rcorless
From: rcorless@uwovax.uwo.ca
Newsgroups: comp.sys.handhelds
Subject: Re: A strange feature (a bug?)
Message-ID: <7367.27133776@uwovax.uwo.ca>
Date: 10 Oct 90 19:00:05 GMT
Lines: 78

Povl Pedersen writes:

>Subject:A strange feature (a bug?)

>There are some problems using differentation/integration when in degrees
>mode on the hp48sx. The manual says in many places that you should make
>sure to switch to RADians mode if appropriate, but If I end up putting
>degrees in the formulae then there is no reason why I should use radians
>when calculating the derived function.


There is an _* excellent reason *_.  The reason is that

          d
          -- sin(x) = cos(x)
          dx

is true _* ONLY IF *_ x is measured in radians.  

You can go back to the definition of derivative to see this.  
(You can tell it is a math prof talking, can't you!).

At x=0, for example, the definition of the derivative of sin(x)
is 

        lim     sin(h) - sin(0)     lim    sin(h)
        h->0    ---------------   = h->0   ------
                      h                      h

and IF h is measured in radians, you can show this limit is 1.

(Incidentally, numerical experiments with limits are an excellent
use of the HP28/48 in class or for the student on his/her own).

If h is measured in degrees, you will find that this limit is
not 1, but rather some strange number (which turns out to be
pi/180).  Just put your calculator in degree mode, and enter
'SIN(H)/H' in the solver, and try a few values of H and see
what you get.  For H very small,  you should get almost exactly
pi/180.


This point is a real puzzle for most students and indeed it
isn't as easy as most profs make out, either, because
essentially we are using the same name "sin" for two different
functions - one where sin(pi) = 0 and another where sin(pi) is
about 0.0548.  This causes confusion.

This question is another candidate for the FAQ list, especially
as the troubleshooting guide in the 48 manual they muck it up
nicely, as follows:  on p 658 in their answers to common questions
they have

Q: When I take the sine of pi in Degrees mode, why do I get
   'SIN(pi)' instead of a number?

and the answer given addresses the issue of "symbolic results
mode" rather than the real answer "Why would you ever want to
take the sine of pi in degree mode?  In degrees, pi is not 
special and you should be taking the sine of 3-and-a-bit
degrees if you want the sine of 3-and-a-bit degrees".

Much the same comment applies with derivatives.  The formulas
for the derivatives of trig functions that everybody learns
are valid only for radian measure.  The HP28/48 cleverly gets
the right formula in the case of degrees, and this is a good
starting point for a discussion of the differences and the
definition of derivative.
-- 
========
Robert Corless, Applied Mathematics, University of Western Ontario
========               London, Ontario, Canada N6A 5B9
e-mail    : RCORLESS@uwovax.uwo.ca
-- 
========
Robert Corless, Applied Mathematics, University of Western Ontario
========               London, Ontario, Canada N6A 5B9
e-mail    : RCORLESS@uwovax.uwo.ca