Re: Simplify for ca^2+sa^2==1

• To: mathgroup at smc.vnet.net
• Subject: [mg26352] Re: [mg26339] Simplify for ca^2+sa^2==1
• From: Andrzej Kozlowski <andrzej at bekkoame.ne.jp>
• Date: Wed, 13 Dec 2000 02:41:14 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```I am frequently discouraged from even attempting to answer questions like
this one because they consist of generalities without any concrete example.
As it is I have to make a guess, whcih may not be right. Anyway,  do you
mean something like this:

In[1]:=
Simplify[c^4*s^2 + 2c^2*s^4 + s^6 + c*s, {c^2 + s^2 == 1}]

Out[1]=
s (c + s)

?

There are other ways to get equivalent answers, which may sometime be more
suitable than th eone given by SImplify. In the above case you could, for
example, do:

In[12]:=
PolynomialReduce[
c^4*s^2 + 2c^2*s^4 + s^6 + c*s, {c^2 + s^2 - 1}, {c, s}][[2]]

Out[12]=
2
c s + s

Which is the same as before, or

In[13]:=
PolynomialReduce[
c^4*s^2 + 2c^2*s^4 + s^6 + c*s, {c^2 + s^2 - 1}, {s, c}][[2]]

Out[13]=
2
1 - c  + c s

and there are other ways (all of them based on GroebnerBasis). I don't think
there is a single best approach, which one to choose will depend largely on
your example and what sort of output you are trying to get.

--
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/

on 12/12/00 4:54 PM, Adalbert Hanssen at hanssen at Zeiss.de wrote:

> Hi, MathGroup,
>
> in a lengthy expression, I know, a lot
> of simplification can be done, if Simplify
> and the like would take into account that
> for varaibles ca and sa
>
> ca^2+sa^2==1
>
> I know, that I can set ca=Sqrt[1-sa^2] and
> deal with the branch cut by hand.
>
> The bad thing is, that these ca^2 and sa^2
> are expanded out in lenghty subexpressions
> involving lots of other symbols. So far, I
> have found no way (but would be glad, if
> someone could advise me one), to factor out
> (ca^2+sa^2).
>
> Unfortunately, there are also terms, where
> (1+ca^2+sa^2) or (2*ca^2+2*sa^2) can be
> factored out, also others with (-ca^2-sa^2)
> and so on.
>
> Any general tip, how to best cope with such
> algebraic manipulations?
>
>
> kind regards
>