Re: cos(12 degress)*cos(24 degrees)*cos(48 degrees)*cos(96 degrees)

On Apr 25, 10:54 pm, "Brad Cooper" <Brad.Cooper...@xxxxxxxxxxx>
writes:

Hi Brad,

Great to get a nice instructive message from you! :)

BC> In Australia VB is famous. Take a look at
BC> http://www.fostersgroup.com/enjoy/beer/victoria_bitter.htm :-)

I believe you the Australians are quite peaceful folks and will
not sue me over non-intended copyright infringement on part
of my parents ;)

But to get serious, I got much interested about your link! Never
expected this. Hope to savour this wonder one fine day!

BC> So, an explicit expression for cos(PI/15) can be constructed
BC> by MuPAD from built-in explicit expressions.

BC> Explicit expressions are returned for the following arguments
BC> (for sin and cos):
BC>
BC> 0, PI/2, PI/3, PI/4, PI/5, 2*PI/5, PI/6, PI/8, 3*PI/8, PI/10,
BC> 3*PI/10, PI/12, 5*PI/12
BC>
BC> I don't think there is a way to do this for cos(PI/31). Maybe
BC> this is why MuPAD gets no further with this one.

A beautiful theory! To verify it experimentally, I entered into
Maple

convert(cos(Pi/31),radical);
cos(1/31*Pi)

like in MuPAD... then I tried

product(cos(Pi*2^j/31), j=0..4):
subs(Pi = 31*X, %):
expand(%):
subs(X = Pi/31, %):
simplify(%);

and got -1/32 ...

Any further comments (from SciFace?) ?

Cheers,

Vladimir


Hi Vladimir,

"Vladimir Bondarenko" <v...@xxxxxxxxxxxxxxx> wrote in message

news:1177552397.747847.317290@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Brad Cooper writes:

BC>> cos(PI/15)*cos(2*PI/15)*cos(4*PI/15)*cos(8*PI/15)

Hello Mr. Cooper,

Enjoy!

Your question is answered by a top SciFace figure.

As you know very well, Dr Stefan Wehmeier is a
MuPAD guru, and within SciFace maybe only a word
of Prof Dr Walter Oevel could weigh more.

I am not quite sure what to say. I do feel very privileged. Of course, you
are also a very exalted person. In Australia VB is famous. Take a look athttp://www.fostersgroup.com/enjoy/beer/victoria_bitter.htm:-)

More below...





On Apr 24, 7:27 am, Stefan Wehmeier <stef...@xxxxxxxxxxx> writes:

SW> You need a little trick:

Hello,

Nice to meet you here!

Your trick does work.

Now a little-devil-in-the-box jumps out and asks you,
what about a little modification of Mr. Cooper's case:

product(cos(PI*2^j/31), j=0..4):

?

In this case,

subs(%, PI = 31*X) :
expand(%):
subs(%, X = PI/31):
simplify(%):
factor(%); // for compactness

1/16*(cos(1/31*PI) - cos(2/31*PI) + cos(3/31*PI)
- cos(4/31*PI) + cos(5/31*PI) - cos(6/31*PI)
+ cos(7/31*PI) - cos(8/31*PI) + cos(9/31*PI)
- cos(10/31*PI) + cos(11/31*PI) - cos(12/31*PI)
+ cos(13/31*PI) - cos(14/31*PI) + cos(15/31*PI) - 1)

... now where is our -1/32 ? ;)

In the MuPAD docs it says...

Explicit expressions are returned for the following arguments (for sin and
cos):

0, PI/2, PI/3, PI/4, PI/5, 2*PI/5, PI/6, PI/8, 3*PI/8, PI/10, 3*PI/10,
PI/12, 5*PI/12

Now,

cos(PI/15) = cos(PI/6-PI/10) = cos(PI/6)*cos(PI/10) + sin(PI/6)*sin(PI/10)

So, an explicit expression for cos(PI/15) can be constructed by MuPAD from
built-in explicit expressions.

I don't think there is a way to do this for cos(PI/31). Maybe this is why
MuPAD gets no further with this one.

Just speculating.

Cheers,

Brad

.



Mit freundlichen Gruessen
Vladimir Bondarenko
Cyber Tester

Brad Cooper wrote:
I thought to use MuPAD 4.0 to calculate the exact value of

cos(12 degress)*cos(24 degrees)*cos(48 degrees)*cos(96 degrees)

i.e. cos(PI/15)*cos(2*PI/15)*cos(4*PI/15)*cos(8*PI/15)

I used...

rewrite(cos(PI/15)*cos(2*PI/15)*cos(4*PI/15)*cos(8*PI/15),exp)
expand(%)
simplify(%)

and other combinations along those lines, but could not get MuPAD to do
the calculation.

You need a little trick:

do
subs(%, PI = 15*X);
expand(%);
subs(%, X = PI/15);
simplify(%);

and everything will work.

--
Stefan Wehmeier
stef...@xxxxxxxxx Hide quoted text -

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