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Re: Inverse function solution

• To: mathgroup at smc.vnet.net
• Subject: [mg132651] Re: Inverse function solution
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Tue, 29 Apr 2014 01:32:50 -0400 (EDT)
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• References: <20140428014444.9FBD16A4E@smc.vnet.net>

$Version


"9.0 for Mac OS X x86 (64-bit) (January 24, 2013)"


sol = Assuming[
  {-1 <= x <= 1, -1 <= y <= 1, C[1] == 0, C[2] == 0},
  Solve[{x == Cos[u], y == Cos[u + v]}, {u, v}] //
   Simplify]


{{u -> ArcTan[x, -Sqrt[1 - x^2]],
     v -> ArcTan[Sqrt[1 - x^2]*(x*y -
              Sqrt[(-1 + x^2)*(-1 + y^2)]),
         y - x^2*y + x*Sqrt[(-1 + x^2)*(-1 + y^2)]]},
   {u -> ArcTan[x, -Sqrt[1 - x^2]],
     v -> ArcTan[Sqrt[1 - x^2]*(x*y +
              Sqrt[(-1 + x^2)*(-1 + y^2)]),
         y - x^2*y - x*Sqrt[(-1 + x^2)*(-1 + y^2)]]},
   {u -> ArcTan[x, Sqrt[1 - x^2]],
     v -> ArcTan[x*y - Sqrt[(-1 + x^2)*(-1 + y^2)],
         -((y - x^2*y + x*Sqrt[(-1 + x^2)*(-1 + y^2)])/
              Sqrt[1 - x^2])]}, {u -> ArcTan[x, Sqrt[1 - x^2]],
     v -> ArcTan[Sqrt[1 - x^2]*(x*y +
              Sqrt[(-1 + x^2)*(-1 + y^2)]), (-1 + x^2)*y +
           x*Sqrt[(-1 + x^2)*(-1 + y^2)]]}}



Bob Hanlon




On Sun, Apr 27, 2014 at 9:44 PM, Narasimham <mathma18 at gmail.com> wrote:

> Solve[ {x == Cos[u], y == Cos[u + v] }, {u, v} ]
>
> Its closed/analytic solution is not possible, even numerically.
>
> The known solutions are ellipses from sine waves with a phase difference,
> having x^2, x y and y^2 terms, as also sketched in Lissajous curves:
>
> ParametricPlot[{Cos[u], Cos[u + v]}, {u, -Pi, Pi}, {v, -Pi, Pi}]
>
> Can there be a work around?
>
> Narasimham
>
>

• References:
  • Inverse function solution
    • From: Narasimham <mathma18@gmail.com>