is hp-49 bad? Or can I not use it correct?
Martin Nielsen
tired of my calc.
I have tried to solve the equation cos(x)^2=sin(2x). It takes no time to
solve it on paper, but I can’t solve it easily on my calc. I have tried to
solve it on the command line by pressing solvex but I get the massage
“error: Not reducible to a rational expression” can that be right? Next I
try to do some trig commands. The only thing which helps is when I press
TEXPAD and then HALFTAN then the calc can solve it, but can it be right,
that the equation have to be expressed by TAN before I can solve it? I can’t
find any logic why I have to do so.
Is it just as difficult on TI-89?
I have also tried to use the num.solver, and it’s okay, but I only get one
solution. Then I have to make another guess. Is it possible to get more
solutions for example by specifying an interval (0 to 2pi) for the variable?
Hope you can help me before I get very angry at my hp-49
Thanks in advance
Martin
Virgil
<***m.ni...@mail1.stofanet.dk> wrote:
[snip]
>
>I have also tried to use the num.solver, and it’s okay, but I only get one
>solution. Then I have to make another guess. Is it possible to get more
>solutions for example by specifying an interval (0 to 2pi) for the
>variable?
>
>Hope you can help me before I get very angry at my hp-49
>
It is in the nature of numeric solvers to find only one solution at a
time. If you specify an interval as a list, e.g., {2 4}, then the
numeric solver will attempt to find a solution in that interval.
--
Virgil
vm...@frii.com
Veli-Pekka Nousiainen
you can use the plotter to pick the root of your interest.
VPN
"Virgil" <vm...@frii.com> wrote in message
news:vmhjr-A3461B....@news.frii.com...
Ivan Cibrario Bertolotti
Martin Nielsen wrote:
>
> [snip]
>
> I have tried to solve the equation cos(x)^2=sin(2x). It takes no time to
> solve it on paper, but I can?t solve it easily on my calc. I have tried to
> solve it on the command line by pressing solvex but I get the massage
> ?error: Not reducible to a rational expression? can that be right? Next I
> try to do some trig commands. The only thing which helps is when I press
> TEXPAD and then HALFTAN then the calc can solve it, but can it be right,
> that the equation have to be expressed by TAN before I can solve it? I can?t
> find any logic why I have to do so.
Granted that I really see nothing wrong in expressing
both sin(x) and cos(x) in terms of tan(x/2) to solve this
equation (it is even the recommended solving method given
on one of the textbooks I have at hand), you can also follow
another methos, similar to the method you would probably
use to solve the equation by hand; it is more cumbersome
to type on the calculator, but it is also more instructive
and let you keep more control on what the calculator is
actually doing.
In the following, for each action you would do
by hand, the corresponding calculator command are listed,
assuming RPN, Real, Exact modes are enabled:
01- enter the equation:
'COS(X)^2=SIN(2*X)'
02- copy the equation and split it to operate on its members:
DUP EXLR
03- linearize the left member and square it:
SWAP TLIN SQ
04- square the right member and express SIN(2*X)^2 in terms of COS:
SWAP SQ TRIGCOS
05- recreate the (squared) equation, substitute 'COS(2*X)=Y',
solve with respect to the auxiliary variable Y and
back substitute:
=
'COS(2*X)=Y' DUP EXLR SWAP = ROT ROT SUBST
'Y' SOLVE SWAP SUBST
'X' SOLVE
06- flatten the list of solution
<< + >> STREAM
07- Now you have the original equation in stack level 2 and
the list of solutions on stack level 1. However, since
these are the solutions of the squared equation, some
of them could very well be invalid. To check which
solutions are valid, back substitute them in the original
equation, letting the arbitrary constant 'n1' be zero.
DUP2 SUBST 'n1=0' SUBST << EVAL >> MAP
In stack level 1 you now have the list of results obtained
back substituting the four potential solutions into the
original equation. As you can see, the second, third
and fourth solutions are valid. The first is not,
because 'COS(ACOS(3/5)/2)^2=-4/5' is clearly not true.
> Is it just as difficult on TI-89?
I am not a TI-89 expert; however, since I own a TI-89, I made some
trials and obtained the following results with AMS 2.05:
a- The 'vanilla' solve(...) command in EXACT mode is unable to solve
the equation; in AUTO mode, it reverts to the numeric solver
and it finds 11 approximate solutions, but no general solution.
b- solve(tExpand(...)) returns the general solution of the
equation.
Notice that the non-trivial solution returned by the TI-89
in b) is expressed in terms of atan(1/2) so it is
very likely that the TI-89 used HALFTAN internally...
nothing new under the sun, indeed.
Hope this helps; with my best regards,
Ivan
--
------------------------------------------------------------------------
Ivan Cibrario Bertolotti / phone: +39-011-3919246
IRITI - National Research Council / fax: +39-011-341882
Turin (Italy) / (*) mailto:cibr...@iriti.cnr.it
(*) The obsolete email address: mailto:cibr...@cstv.to.cnr.it will be
dismissed soon; please update your address books
Parisse Bernard
> trials and obtained the following results with AMS 2.05:
>
> a- The 'vanilla' solve(...) command in EXACT mode is unable to solve
> the equation; in AUTO mode, it reverts to the numeric solver
> and it finds 11 approximate solutions, but no general solution.
>
> b- solve(tExpand(...)) returns the general solution of the
> equation.
>
> Notice that the non-trivial solution returned by the TI-89
> in b) is expressed in terms of atan(1/2) so it is
> very likely that the TI-89 used HALFTAN internally...
> nothing new under the sun, indeed.
>
I believe once you have done texpand on the equation, the TI does
factor then tcollect. This would explain atan(1/2) and why it does not
solve it without user intervention (seems strange that you need to do
tcollect after texpand). I have decided to change a little bit
the rewriting rules before solve so that HALFTAN is called after
TEXPAND (already called) in this case.
John H Meyers
> I have tried to use the num.solver, and it’s okay
When I first saw it (on the HP34C), I thought it was fantastic,
but now, even quite improved, it is only "okay"; that's what
20 years of routine availability has done to a former miracle :)
> but I only get one solution. Then I have to make another guess.
> Is it possible to get more solutions for example by specifying
> an interval (0 to 2pi) for the variable?
Yes, using a small program to help you; note, however, that this
still leaves to you a judgment about how "coarse" or "fine"
to make the search for roots over the interval. There may be some
more analytic symbolic tools which endeavor to predict locations
of roots, but the basic *numeric* solver (and integrator) are
still just number-crunchers; they do not analyze the function,
they just try some inputs and see what comes out, and then
"sample" the "black box" function some more. Your own
intelligent guidance on where to start and how throughly to "dig"
are therefore still necessary, just as in many other not yet
fully automated tasks in the living of life.
A "ROOTS" program (scan for roots over an interval), which may
help you with your quest, may be found by following this link:
http://www.deja.com/getdoc.xp?AN=632515576&fmt=text
> Hope you can help me before I get very angry at my hp-49
Change the approach to the wonder of new discovery about to be made,
and the joy of personal involvement in the pursuit of knowledge;
then the journey will be much more pleasant.
That's why we have http://www.mum.edu, where the whole student body
*loves* their education, which ought to be how it is everywhere, no?
-----------------------------------------------------------
With best wishes from: John H Meyers <jhme...@mum.edu>
Sent via Deja.com http://www.deja.com/
Before you buy.
Veli-Pekka Nousiainen
(again)
How complicated must rules behind the scenes be?
(a rhetoric question)
VPN
"Parisse Bernard" <par...@fourier.ujf-grenoble.fr> wrote in message
news:39899522...@fourier.ujf-grenoble.fr...
Veli-Pekka Nousiainen
news:8mdo4a$56c$1...@nnrp1.deja.com...
> Martin Nielsen:
>
> > I have tried to use the num.solver, and it's okay
> > Is it possible to get more solutions for example by specifying
> > an interval (0 to 2pi) for the variable?
> A "ROOTS" program (scan for roots over an interval), which may
> help you with your quest, may be found by following this link:
>
> http://www.deja.com/getdoc.xp?AN=632515576&fmt=text
>
http://165.234.32.14/hp49/math/numeric/ss49v12.zip
http://165.234.32.14/hp49/math/numeric/ss49docs.zip
VPN
Ivan Cibrario Bertolotti
Parisse Bernard wrote:
>
> I believe once you have done texpand on the equation, the TI does
> factor then tcollect. This would explain atan(1/2) and why it does not
> solve it without user intervention (seems strange that you need to do
> tcollect after texpand). I have decided to change a little bit
> the rewriting rules before solve so that HALFTAN is called after
> TEXPAND (already called) in this case.
You are right, my fault.
If you linearize cos(x)^2 as (1 + cos(2x)/2) (TLIN),
and then express everything is terms of tan(x) (HALFTAN),
and then SOLVE, the solution will contain atan(1/2);
If you express sin(2x) as 2sin(x)cos(x) (TEXPAND),
express everything in terms of tan(x/2) (HALFTAN), and
then SOLVE, the solution will contain -+2arctan(+-2+sqrt(5)).
Since I manually forced tExpand() first on the 89,
the only explanation of atan(1/2) is an implicit tCollect()
made by solve().
A little question: in your opinion would it be
possible/useful to make TLIN and possibly all other rewrite
commands operate on both sides of an *equation*?
TEXPAND, for example, already works this way, but
'cos(x)^2=sin(2x)' TLIN does nothing instead. Both commands
work on *expressions*, of course.
In my opinion, this would reduce the need of EXLR
when rewriting an equation outside the eqw, when you have
no means to select a subequation to process.
Thank you very much; best regards,
Parisse Bernard
for this. I could fix it.