Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2006
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2006

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Re: Re: simplify a trig expression

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65470] Re: [mg65449] Re: [mg65436] Re: [mg65415] simplify a trig expression
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Mon, 3 Apr 2006 06:59:37 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200603311109.GAA15029@smc.vnet.net> <200604011038.FAA07301@smc.vnet.net> <200604020900.FAA01612@smc.vnet.net> <11D40ADD-9EC9-4DCE-B685-1CA00605B9B2@mimuw.edu.pl>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

Actually, what I was trying to do is this:  To obtain in Mathematica, 
the answers to a ten-question integration exam that would be of the form 
students would obtain with standard paper-and-pencil techniques.  And 
the purpose of that was to to provide to the graders, whom I supervise, 
answers that are unquestionably correct -- and, again, in that form.

Andrzej Kozlowski wrote:
> First, let me state a fundamental objection to the whole enterprise. I 
> am of the opinion that there is on the whole very little point in 
> putting a lot of effort into forcing computer algebra programs return 
> the answers you already know to be correct. The purpose of such programs 
> is to simplify expressions that are too complicated too analyse by hand 
> so as to make them simple enough to handle by hand. The "handling by 
> hand" part is precisely what computer programs are not very good at and 
> usually it takes longer to make them do so then just typing in the 
> answer. Of course once you already know the answer it is usually very 
> easy to verify that it equal the one given by Mathematica. For example, 
> in your situation the best approach, in my opinion would be simply this:
> 
> 
> FullSimplify[D[Integrate[Sin[x]/(1 - Cos[x]), x] -
>     Log[1 - Cos[x]], x]]
> 
> 
> 0
> 
> As far as I can see this is all you would even need to do in practice. 
> The rest is just playing with CAS. But if you like playing then by using 
> FullSimplify with suitable TransformationFunctions and 
> ComplexityFunction you can usually transform anything into anything 
> else. So in this case you can do, for example,
> 
> 
> f[(n_)*Log[a_]] := Log[a^n];
> f[Cos[x_]^2] :=  (1/2)*(Cos[2*x] + 1);
> f[Sin[2*(x_)]^2] := (1/2)*(1 - Cos[x]);
> 
> 
> 
> FullSimplify[Integrate[Sin[x]/(1 - Cos[x]), x],
>   TransformationFunctions -> {Automatic, f},
>   ComplexityFunction ->
>    (LeafCount[#1] + 50*Count[#1, x/2, {0, Infinity}] +
>      Count[#1, _Power, {0, Infinity}, Heads -> True] & )]
> 
> 
> Log[(1/2)*(1 - Cos[x])]
> 
> It is slightly amusing to analyse exactly why this works, but I 
> amusement is about all of value I can see in this aproach.
> 
> Andrzej Kozlowski
> 
> On 2 Apr 2006, at 11:00, Murray Eisenberg wrote:
> 
>> OK, now how about the following--another case where Mathematica gives a
>> more complicated looking answer than the typical paper-and-pencil direct
>> substitution would provide?
>>
>> Mathematica gives:
>>
>>    Integrate[Sin[x]/(1 - Cos[x]), x]
>> 2*Log[Sin[x/2]]
>>
>> I'd like an answer in the form of Log[1-Cos[x]]  (plus a constant, to
>> actually equal the above).  The best I've been able to do so far is:
>>
>>    (Integrate[Sin[x]/(1 - Cos[x]), x]
>>       // Simplify[# /. a_*Log[b_]:>Log[b^a]] &
>>       //MapAt[TrigReduce, #, 1]&) /. (Log[c_ b_]->Log[b]+Log[c])
>> -Log[2] + Log[1 - Cos[x]]
>>
>> Is there some easier way?
>>
>> Andrzej Kozlowski wrote:
>>> This is one of those cases where FullSimplify will not work because
>>> it lacks a suitable transformation function. In this particular case
>>> the transformation function is of the form:
>>>
>>> f[n_*Log[a_]] := Log[a^n]
>>>
>>> Of course this is only valid with various assumptions on n and a, but
>>> I won't bother with this here. Anyway, observe that:
>>>
>>>
>>> FullSimplify[Integrate[Cos[x]/(Sin[x] + 1), x],
>>>    TransformationFunctions -> {Automatic, f}]
>>>
>>>
>>> Log[Sin[x] + 1]
>>>
>>> Note also that Simplify will not work even when you add f.
>>>
>>> I am not sure if there are good reasons for adding a version of f
>>> (taking account of suitable assumptions) to the default
>>> transformation functions of FullSimplify. It may however be a good
>>> idea to have another possible value for the option
>>> TransformationFunctions besides only Automatic and user defined ones.
>>> In fact I have suggested in the past one or two other useful
>>> TransformationFunctions; perhaps it might be a good idea to define
>>> more and  collect them into a single option value or maybe several.
>>>
>>> Andrzej Kozlowski
>>>
>>>
>>>
>>>
>>>
>>>
>>> On 31 Mar 2006, at 13:09, Murray Eisenberg wrote:
>>>
>>>> A direct substitution (with paper and pencil) gives that the
>>>> integral of
>>>>   Cos[x]/(Sin[x] + 1) is Log[Sin[x] + 1].  This is valid provided
>>>> Sin[x]
>>>> is not -1.
>>>>
>>>> Mathematica gives:
>>>>
>>>>    Integrate[Cos[x]/(Sin[x] + 1), x]
>>>> 2 Log[Cos[x/2] + Sin[x/2]]
>>>>
>>>> Is there some simple way to coerce the latter Mathematica-supplied
>>>> result into the paper-and-pencil answer?
>>>>
>>>> The closest I could get is:
>>>>
>>>>    Log[TrigExpand[Expand[(Cos[x/2] + Sin[x/2])^2]]] /.
>>>>    {Sin[x/2] -> Sqrt[(1 - Cos[x])/2],
>>>>     Cos[x/2] -> Sqrt[(1 + Cos[x])/2]}
>>>> Log[1 + Sqrt[1 - Cos[x]]*Sqrt[1 + Cos[x]]]
>>>>
>>>> Am I not seeing some easier TrigExpand or TrigReduce method?
>>>>
>>>> --Murray Eisenberg                     murray at math.umass.edu
>>>> Mathematics & Statistics Dept.
>>>> Lederle Graduate Research Tower      phone 413 549-1020 (H)
>>>> University of Massachusetts                413 545-2859 (W)
>>>> 710 North Pleasant Street            fax   413 545-1801
>>>> Amherst, MA 01003-9305
>>>>
>>>
>>>
>>
>> --Murray Eisenberg                     murray at math.umass.edu
>> Mathematics & Statistics Dept.
>> Lederle Graduate Research Tower      phone 413 549-1020 (H)
>> University of Massachusetts                413 545-2859 (W)
>> 710 North Pleasant Street            fax   413 545-1801
>> Amherst, MA 01003-9305
>>
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


  • Prev by Date: Re: Re: Re: simplify a trig expression
  • Next by Date: HoldFirst confusion
  • Previous by thread: Re: Re: Re: simplify a trig expression
  • Next by thread: Re: simplify a trig expression