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Re: Assumptions for Trigonometry Inequalities

• To: mathgroup at smc.vnet.net
• Subject: [mg71391] Re: [mg71328] Assumptions for Trigonometry Inequalities
• From: Adam Strzebonski <adams at wolfram.com>
• Date: Thu, 16 Nov 2006 00:53:57 -0500 (EST)
• References: <200611151143.GAA01230@smc.vnet.net> <ADA44405-975D-4BFA-8D35-FAEC708342D2@mimuw.edu.pl>
• Reply-to: adams at wolfram.com

The assumptions mechanism used by Simplify generally can only prove
polynomial inequalities. It can prove a very limited number of
transcendental inequalities that follow (through a polynomial
dependence) from a collection of hard-coded inequalities.

Inequalities

In[5]:= Simplify[Sin[gamma]>0,0<gamma<Pi]
Out[5]= True

In[6]:= Simplify[Sin[gamma]<0,Pi<gamma<2Pi]
Out[6]= True

are hard coded in Simplify, inequalities for negative gamma are not.

Reduce uses more general inequality solving methods (but has too high
complexity to be used automatically by the assumption mechanism).

In[9]:= Reduce[-Pi<gamma<0 && Sin[gamma]>=0, gamma]
Out[9]= False

Best Regards,

Adam Strzebonski
Wolfram Research

Andrzej Kozlowski wrote:
> 
> On 15 Nov 2006, at 20:43, Martin Schoenecker wrote:
> 
>> Hello,
>>
>> the Sin of a real number between zero and Pi should be positive, as
>> Simplify finds out, too.
>>
>> In[133]:= Simplify[Sin[gamma]>0,0<gamma<Pi]
>> Out[133]= True
>>
>> As well, the Sin of a number between -Pi and 0 is negative, but why
>> doesn't Simplify evaluate it?  The Less implies that the variable is
>> real, I think, however stating it explicitly doesn't change the result.
>>
>> In[134]:= Simplify[Sin[gamma]<0,-Pi<gamma<0]
>> Out[134]= Sin[gamma]<0
>>
>> Are there any hints on that?
>> Thank you,
>> Martin
>>
> 
> I think these kind of inequalities can't be determined by "general 
> methods" available to Simplify, so it just needs to "know them". In this 
> case it appears that there is a "gap in its knowledge". Note however 
> that if you replace your statement by its equivalent (using gamma -> 
> Pi/2-delta), you get the right answer:
> 
> 
> Simplify[Cos[delta] < 0, Pi/2 < delta < (3*Pi)/2]
> 
> True
> 
> Andrzej Kozlowski
> 
> 

• References:
  • Assumptions for Trigonometry Inequalities
    • From: Martin Schoenecker <ms_usenet@gmx.de>