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MathGroup Archive 2005

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Re: inequations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg62847] Re: inequations
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 6 Dec 2005 23:10:17 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <dn368i$2ri$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Norbe wrote:
> Can somebody help me with this Inequation?
> 
> (( 2*Sin [a] ) / (3*Pi / 4 ) -a ) +1 -d  >= 0
> 
> I need solve for a, but i can't found any function or easy form for solve.
> 
> 
> Thank
> 
Hi,

An obvious choice would be the function *Reduce*, unfortunately it fails 
to solve the inequality:

In[1]:=
expr = ((2*Sin[a])/(3*(Pi/4)) - a) + 1 - d;

In[2]:=
Reduce[expr >= 0, a]

Reduce::nsmet: This system cannot be solved with the methods available 
to Reduce. More...

Out[2]=
Reduce[1 - a - d + (8*Sin[a])/(3*Pi) >= 0, a]

Therefore, you might be on your own to find a general formula. Studying 
some particular cases by giving some values to the parameter d and 
plating the graph of the functions f(a) = 1 - a - d + (8*Sin[a])/(3*Pi) 
and y = d should help visualizing what's going on (f{a} is decreasing on 
R and we are interested in the intersection point of f(a) and the 
horizontal straight line y = d. Then, the required value of a are on the 
left of the graph). You could try

In[3]:=
Plot[{expr /. d -> 1, 1}, {a, -10, 10}];

or the following one that gives a family of intersecting curves and 
their intersection points:

In[4]:=
Needs["Graphics`Graphics`"]
Plot[Evaluate[Table[{expr /. d -> n, n}, {n, -5, 5}]], {a, -10, 10},
    PlotStyle -> {{Red}, {Blue, Dashing[{0.015}]}}, ImageSize -> 500,
    Frame -> True, FrameTicks -> {UnitScale[-3*Pi, 3*Pi, 8/(3*Pi)],
      Automatic, None, None}, Axes -> None,
    Epilog -> {PointSize[0.015], Point /@
       Table[{a /. FindRoot[expr == n /. d -> n, {a, -n}], n},
        {n, -5, 5}]}];

Hope this helps,
/J.M.


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