# [Maxima] Simplifying atan expressions

Stavros Macrakis macrakis at alum.mit.edu
Mon Feb 20 15:12:33 CST 2012

```Henry,

Yes, this derivation is essentially the same as the bottomup rectform.  In
fact:

expr: logcontract(logarc(4*atan(1/5)-atan(1/239)));
rectform(substpart(e1:rectform(piece),expr,1,2,1))  => %pi/4

You'll see that e1 here is -%i.

What rectform does internally by default looks more like

rectform(substpart(e2:polarform(piece),expr,1,2,1));

where e2 isn't at all helpful.

I agree that most mathematical functions (sin, etc.) and form-changing
routines (factor etc.) should distribute over "=".

-s

On Mon, Feb 20, 2012 at 15:45, Henry Baker <hbaker1 at pipeline.com> wrote:

> result=4*atan(1/5)-atan(1/239);
>                                        1          1
> (%o1)                   result = 4 atan(-) - atan(---)
>                                        5         239
> (%i2) lhs(%)=logcontract(logarc(rhs(%)));
>                                              4
>                                      (%i - 5)  (%i + 239)
>                             %i log(- --------------------)
>                                                         4
>                                      (%i - 239) (%i + 5)
> (%o2)               result = ------------------------------
>                                           2
> (%i3) %*2/%i;
>                                                4
>                                        (%i - 5)  (%i + 239)
> (%o3)             - 2 %i result = log(- --------------------)
>                                                           4
>                                        (%i - 239) (%i + 5)
> (%i4) exp(lhs(%))=exp(rhs(%));
>                                               4
>                     - 2 %i result     (%i - 5)  (%i + 239)
> (%o4)              %e              = - --------------------
>                                                          4
>                                       (%i - 239) (%i + 5)
> (%i5) lhs(%)=rectform(rhs(%));
>                              - 2 %i result
> (%o5)                       %e              = - %i
> (%i6) log(lhs(%))=log(rhs(%));
> (%o6)                      - 2 %i result = log(- %i)
> (%i7) rectform(lhs(%))=rectform(rhs(%));
>                                             %i %pi
> (%o7)                      - 2 %i result = - ------
>                                               2
> (%i8) solve(%,result);
>                                          %pi
> (%o8)                           [result = ---]
>                                           4
>
> This example shows why rectform, log, exp, logcontract, logarc, etc.,
> should distribute over "=".
>
> At 11:20 AM 2/20/2012, Stavros Macrakis wrote:
> >Consider the expression 4*atan(1/5)-atan(1/239) (Machin's formula).
> >
> >How can we use Maxima to simplify it to %pi/4?
> >
> >Well, here's one approach:
> >
> >atan(trigexpand(trigexpand(tan(ex))))
> >
> >And here's another way that doesn't depend on trig identities:
> >
> >(%i1) ex:4*atan(1/5)-atan(1/239);
> >(%o1) 4*atan(1/5)-atan(1/239)
> >(%i2) llex: logcontract(logarc(ex));
> >(%o2) %i*log(-(%i-5)^4*(%i+239)/((%i-239)*(%i+5)^4))/2
> >(%i3) rectform(llex);
> >(%o3)
> (atan(sin(4*atan(1/5))/cos(4*atan(1/5)))-atan(sin(4*(%pi-atan(1/5)))/cos(4*(%pi-atan(1/5))))-2*atan(1/239))/2
> >        <<< oops, not so helpful
> >(%i4) scanmap(rectform,llex,bottomup);
> >        <<< but reorganizing the rectform calculation gives a better
> result
> >        <<< makes me think that rectform could be improved...
> >(%o4) %pi/4
> >
> >Are any of these techniques packaged into some analog of trigsimp?
> >
> >                -s
>
>
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