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MathGroup Archive 1998

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Re: Trigonometric equation????

  • To: mathgroup at smc.vnet.net
  • Subject: [mg12983] Re: Trigonometric equation????
  • From: Daniel Lichtblau <danl>
  • Date: Sun, 28 Jun 1998 02:52:07 -0400
  • Organization: Wolfram Research, Inc.
  • References: <6mqf9u$3i1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Rikis wrote:
> 
> I'm in trouble: I have to isolate x and y in this system of equations
> and I don't know how...
> 
> Could you please help me?
> 
>     L = C*x  +  (2*R) / tan(y/2)
> 
>     x = 2 * ( (V - R - R/cos(y)) * tan(y) )
> 
> L,C,R,V are constants
> 
> Thank you very much!!!


Playing around a bit I find that the following works. I believe the use
of Together helped to avoid excessive complications caused by adding
auxiliary variables and equations to handle denominators produced by
TrigToExp.

It also helps to have input that is in Mathematica InputForm.

In[12]:= eqs = Together[TrigToExp /@
        {-L + C*x  +  (2*R) / Tan[y/2],
          -x + 2 * ( (V - R - R/Cos[y]) * Tan[y])}];

In[13]:= Timing[sol = Solve[eqs==0, {x,y}];] Solve::ifun: Inverse
functions are being used by Solve, so some solutions may
     not be found.
Out[13]= {3.12 Second, Null}

In[14]:= Length[sol]
Out[14]= 5

In[15]:= LeafCount[sol]
Out[15]= 10021


Playing some more, I find a method that avoids conversion to
exponentials and also gives a simpler result, as measured in LeafCount.

In[58]:= eqs = TrigExpand /@
                {-L + C*x  +  (2*R) / Tan[z],
                  -x + 2 * ( (V - R - R/Cos[2*z]) * Tan[2*z])};

In[59]:= Timing[sol = Solve[eqs==0, {x,z}] /. (z->a_) :> y->2*a;]
Solve::ifun: Inverse functions are being used by Solve, so some
solutions may
     not be found.
Out[59]= {2.01 Second, Null}

In[60]:= Length[sol]
Out[60]= 5

In[61]:= LeafCount[sol]
Out[61]= 706

In[62]:= sol[[1]] // InputForm
Out[62]//InputForm=
{x -> (L - 2*R*Root[-L + 2*R*#1 - 4*C*V*#1 + 2*L*#1^2 - 4*R*#1^3 -
        8*C*R*#1^3 + 4*C*V*#1^3 - L*#1^4 + 2*R*#1^5 & , 1])/C,
 y -> 2*ArcCot[Root[-L + 2*R*#1 - 4*C*V*#1 + 2*L*#1^2 - 4*R*#1^3 -
       8*C*R*#1^3 + 4*C*V*#1^3 - L*#1^4 + 2*R*#1^5 & , 1]]}


Daniel Lichtblau
Wolfram Research


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