Solve returns format I don't understand with dollar sign ($) [closed]

Question

I have the following function

K[theta_, phi_] := Module[{c1, c2, c3, Phi, PhiPr, x, y},
  {c1, c2, c3} = 
   1 - 3 Sin[theta]^2 Cos[phi - 2 Pi (# - 1)/3]^2 & /@ {1, 2, 3};
  Phi = Sqrt[((c2 + c3)^2 - c1^2)/(4 c2 c3)];
  PhiPr = ((c3 - c2)/c1) Phi;
  If[0 <= Phi^2 <= 1,
   {x, y} /. 
    Solve[Sin[Sqrt[3] x/2] == Phi && 
      Sin[3 y/2] == 
       PhiPr && (c2 + c3) Cos[Sqrt[3] x/2] == -c1 Cos[3 y/2] && -Pi <=
        x <= Pi && -Pi <= y <= Pi, {x, y}]
   , Null]
  ]

Sometimes it behaves itself and returns me either a Null (as desired) or an actual number. However sometimes it is a naughty function and returns something strange. For example if I compute K[0.1,0] it returns me {x$17889, y$17889}.

What is this output? I don't understand. Many thanks

Answer 1

You can identify the problem by performing the steps in your function one at a time for theta = .1 and phi = 0..

{c1, c2, c3} = 1 - 3 Sin[0.1]^2 Cos[0 - 2 Pi (# - 1)/3]^2 & /@ {1, 2, 3}
{* {0.9701, 0.992525, 0.992525} *}

Phi = Sqrt[((c2 + c3)^2 - c1^2)/(4 c2 c3)]
{* 0.87245 *}

PhiPr = ((c3 - c2)/c1) Phi
{* 0. *}

Because 0 <= Phi^2 <= 1 is True, the Solve is executed:

{x, y} /. Solve[Sin[Sqrt[3] x/2] == Phi && Sin[3 y/2] == 
    PhiPr && (c2 + c3) Cos[Sqrt[3] x/2] == -c1 Cos[3 y/2] && -Pi <= x <= Pi && -Pi <= y <= Pi, 
    {x, y}]

returning the error message Solve::ratnz and {x, y} instead of an answer. Returning now to your original function, a Module, x and y are temporary variables, which have names beginning with $, which is what you are getting.

Addendum

Replace the If statement in the Module by

If[0 <= Phi^2 <= 1, Check[{x, y} /. Solve[Sin[Sqrt[3] x/2] == Phi && 
  Sin[3 y/2] == PhiPr && (c2 + c3) Cos[Sqrt[3] x/2] == -c1 Cos[3 y/2] && -Pi <= x <= Pi &&
  -Pi <= y <= Pi, {x, y}], Null], Null]]

to obtain a Null response is such cases.