Assumptions for Trigonometry Inequalities

[Martin Schoenecker]

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

[dimitris]

Looks strange because Mathematica evaluates correctly e.g.

Sin /@ Range[-Pi, 0, Pi/6]
{0, -(1/2), -(Sqrt[3]/2), -1, -(Sqrt[3]/2), -(1/2), 0}

Anyway, if you try

Simplify[Sin[gamma] < 0, Pi < gamma < 2*Pi]
True

works fine.

[Andrzej Kozlowski]

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

[Adam Strzebonski]

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