convert/radical
Andrew McLean
convert(sin(Pi/27), 'radical');
it just returned sin(Pi/27). Although "solve(z^27=1,z);" does show the
radical form of the 27th roots of unity, so the conversion is possible. Any
ideas how I should get the conversion of the trig function to work?
Maplesoft tech support were not very helpful, apparently only instalation
support is available for the trial version.
--
Dr Andrew McLean
Marine and Acoustics Centre
QinetiQ ltd
The views expressed above are entirely those of the writer and do
not represent the views, policy or understanding of any other
person or official body.
Robert Israel
Andrew McLean <almc...@taz.qinetiq.com> wrote:
>I have been evaluating the trial version of Maple 8 and I tried
>convert(sin(Pi/27), 'radical');
>it just returned sin(Pi/27). Although "solve(z^27=1,z);" does show the
>radical form of the 27th roots of unity, so the conversion is possible. Any
>ideas how I should get the conversion of the trig function to work?
>Maplesoft tech support were not very helpful, apparently only instalation
>support is available for the trial version.
I think it's more of a mathematical question than a technical-support
question.
There may be a more direct way, but I'd start from the Chebyshev
polynomial T[27]: T[n](cos(t)) = cos(n*t). So cos(Pi/27) is a root
of T[27]+1. Let's factor that, because there are lots of other roots:
> F:= factor(orthopoly[T](27,c)+1);
2 3 2
F := (c + 1) (2 c - 1) (8 c - 6 c - 1)
9 7 5 3 2
(512 c - 1152 c + 864 c - 240 c + 18 c - 1)
Presumably the factor we want is the last one. To check:
> combine(eval(op(4,F),c=cos(Pi/27)),trig);
0
> F:= op([4,1],F);
9 7 5 3
F := 512 c - 1152 c + 864 c - 240 c + 18 c - 1
So this would be a minimal polynomial for cos(Pi/27). Now if
c = cos(Pi/27) and s = sin(Pi/27) we have c^2 + s^2 = 1. Eliminate
the c using "resultant".
> G:= resultant(F,c^2+s^2-1,c);
2 4 6 8 10 12
G := -3 + 324 s - 8640 s + 88704 s - 456192 s + 1317888 s - 2236416 s
14 16 18
+ 2211840 s - 1179648 s + 262144 s
Attempting to factor G, we see that it is irreducible. So it must be
a minimal polynomial for sin(Pi/27). Now we can solve for s.
> S:= [solve(G,s)];
I won't include the voluminous output. Of course there are 18 roots here,
and only one of them will be sin(Pi/27). We can use "evalf" to
approximate each numerically, and pick out the one that is numerically
close to sin(Pi/27).
> select (t -> evalf(abs(t - sin(Pi/27))) < .001, S);
1/3 2/3 1/3 1/2 2/3 1/2 1/2
(-%1 (%1 + 4 - 8 %1 + I 3 %1 - 4 I 3 ))
[1/4 -----------------------------------------------------------]
1/3
%1
1/2 2/3 1/2 1/3 1/2 1/2 1/3
%1 := -(4 + 4 I 3 ) - (4 + 4 I 3 ) + I 3 (4 + 4 I 3 ) + (
1/2 4/3 1/2 2/3 1/2 1/2 2/3
-32 + (4 + 4 I 3 ) - 2 (4 + 4 I 3 ) - 2 I 3 (4 + 4 I 3 )
1/2
)
So there's your answer.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Alec Mihailovs
"Andrew McLean" <almc...@taz.qinetiq.com> wrote in message
news:ad4qlk$j3o$1...@hamble.qinetiq.com...
> I have been evaluating the trial version of Maple 8 and I tried
>
> convert(sin(Pi/27), 'radical');
>
> it just returned sin(Pi/27). Although "solve(z^27=1,z);" does show the
> radical form of the 27th roots of unity, so the conversion is possible.
Any
> ideas how I should get the conversion of the trig function to work?
As it is written in the convert/radical help, if the argument of a
trigonometric
function is of the form n/120*Pi where n is an integer, then Maple converts
the function to radical form. For Pi/27 one can use 2 step conversion:
> convert(convert(sin(Pi/27),RootOf),radical);
The answer is not as good-looking though as one Dr. Robert Israel found.
Best wishes,
Alec Mihailovs
http://webpages.shepherd.edu/amihailo/
Carl Devore
On Fri, 31 May 2002, Alec Mihailovs wrote:
> "Andrew McLean" <almc...@taz.qinetiq.com> wrote in message
> news:ad4qlk$j3o$1...@hamble.qinetiq.com...
> > convert(sin(Pi/27), 'radical');
> > it just returned sin(Pi/27). Although "solve(z^27=1,z);" does show the
> > radical form of the 27th roots of unity, so the conversion is possible.
>
> convert(convert(sin(Pi/27),RootOf),radical);
That is the kind of answer that Maple's technical support should have been
able to give.
Alec Mihailovs
Actually, if such expressions are allowed,
the sin(Pi/27) can be written in much more simple form :-)
> (((1+I*sqrt(3))/2)^(1/9)-((1-I*sqrt(3))/2)^(1/9))/2/I;
W. Edwin Clark
Alec Mihailovs wrote:
>
> As it is written in the convert/radical help, if the argument of a
> trigonometric
> function is of the form n/120*Pi where n is an integer, then Maple converts
> the function to radical form. For Pi/27 one can use 2 step conversion:
>
> > convert(convert(sin(Pi/27),RootOf),radical);
>
Alec,
What version of Maple are you using for this? On my Mac with
Maple 7 the above line just returns sin(Pi/27).
Edwin
------------------------------------------------------------
W. Edwin Clark, Math Dept, University of South Florida,
http://www.math.usf.edu/~eclark/
------------------------------------------------------------
Alec Mihailovs
"W. Edwin Clark" <ecl...@math.usf.edu> wrote in message
news:3CF81894...@math.usf.edu...
> > > convert(convert(sin(Pi/27),RootOf),radical);
> What version of Maple are you using for this? On my Mac with
> Maple 7 the above line just returns sin(Pi/27).
Edwin,
I used the newly arrived Maple 8. It gave a very long answer, a fraction
involving a lot of such items as (4+4*sqrt(3)*I)^(2/3) etc. I just checked
it on
Maple 7 on Windows and it gave me the same answer that you got.
By the way, it seems as though there is no Mac version on the 2 CDs with
Maple 8.
First CD contains Windows and Linux versions, and second one - AIX,
Solaris, Linux network and HP-UX. Windows version looks significally
larger than Maple 7. It contains a new folder with examples and a series
of Java related additions including Maplets.
Best wishes,
Alec
Robert Israel
W. Edwin Clark <ecl...@math.usf.edu> wrote:
>Alec Mihailovs wrote:
>> > convert(convert(sin(Pi/27),RootOf),radical);
>What version of Maple are you using for this? On my Mac with
>Maple 7 the above line just returns sin(Pi/27).
This is a new feature of Maple 8.
Alec Mihailovs
"W. Edwin Clark" <ecl...@math.usf.edu> wrote in message
news:3CF81894...@math.usf.edu...
> > > convert(convert(sin(Pi/27),RootOf),radical);
> >
>
> Alec,
>
> What version of Maple are you using for this? On my Mac with
> Maple 7 the above line just returns sin(Pi/27).
Edwin,
I found out today that convert/RootOf/sincos already existed in Maple 7, it
wasn't just included in convert/RootOf. That means that
> convert(convert(sin(Pi/27),`RootOf/sincos`),radical);
would convert sin(Pi/27) into a radical form in Maple 7 and the following
would give the minimal polynomial and an approximation:
> convert(sin(Pi/27),`RootOf/sincos`);
Comparing
> showstat(`convert/RootOf`);
in Maple 7 and Maple 8 shows that Maple 8 added 2 lines at the beginning:
1 if f::specfunc(algebraic,{'sin', 'cos'}) and (op(f)/Pi)::rational
then
2 `convert/RootOf/sincos`(f)
Best wishes,
Alec
W. Edwin Clark
Alec Mihailovs wrote:
> I found out today that convert/RootOf/sincos already existed in Maple 7, it
> wasn't just included in convert/RootOf. That means that
> > convert(convert(sin(Pi/27),`RootOf/sincos`),radical);
>
> would convert sin(Pi/27) into a radical form in Maple 7 and the following
> would give the minimal polynomial and an approximation:
>
This also works in Maple 6, but not in Maple V.
Edwin