Solving for vector elements in a function to which the vectors are not passed explicitly

Question

I want to define a function which takes in two integers (indicating the lengths of 2 vectors), and solves a simple set of expressions at a set of points to find all the values in both the vectors.

so far I have managed to successfully create a function which does exactly what I say above, except that rather than taking two integers as arguments, it takes the two vectors itself. Forgive me for the length of this next code fragment:

solution[svec_, nvec_] := 
  NSolve[
   Table[
     approx[svec, nvec, -Cos[n*Pi/(Length[nvec] + Length[svec])]] == 
       exact[-Cos[n*Pi/(Length[nvec] + Length[svec])]],
     {n, 1, Length[nvec] + Length[svec]}],
   Join[svec, nvec]];

The

-Cos[n*Pi/(Length[nvec] + Length[svec])

part is just a set of points. All of the above works fine.

My attempt to change it so that it takes in just integers was to do the following:

STEP ONE: create a function which takes an integer, gives a vector of length integer:

nvec[Q_] := Table[Unique["n"], {Q}];
svec[M_] := Table[Unique["s"], {M}];

STEP TWO: replace all of the svecs and nvecs in the previous code fragment with nvec[Q] and svec[M]. (could also get rid of Lengths obviously):

solution[Q_, M_] := 
  NSolve[
    Table[
      approx[svec[M], nvec[Q], -Cos[n*Pi/(Q + M)]] == exact[-Cos[n*Pi/(Q + M)]],
      {n, 1, M + Q}],
    Join[svec[M], nvec[Q]]];

STEP THREE: Call the function

sol = solution[10, 20]

For some reason, this doesn't work. I never get an error; it just never finishes evaluating. Any help would be very much appreciated