Another possible solution is to

1. ContourPlot the equation directly;

2. Make the coordinate transform on the coordinates of points inside the resulting graphic.

I've wrapped these steps in a function:

ClearAll@implicitPlot
implicitPlot[eq_, range__, coordSys_, opt : OptionsPattern[ContourPlot]] := 
 Module[{coord}, 
  With[{trans = #[coord, #2] &[Function, 
       CoordinateTransform[coordSys -> "Cartesian", {coord[1], coord[2]}]] /. 
      coord[i_] :> Part[coord, i]}, 
   Module[{plot = ContourPlot[eq, range, opt, PlotRange -> All]},
    plot /. 
     GraphicsComplex[coord_, rest_] :> 
      GraphicsComplex[trans[coord\[Transpose]]\[Transpose], rest]]]]

implicitPlot[Cos@r == theta, {r, 0, 40}, {theta, -8 Pi, 8 Pi}, "Polar", 
 PlotPoints -> 100]

The advantage of this approach is, it allows us to directly set domain of definition under the interested coordinate system.

Another possible solution is to

1. ContourPlot the equation directly;

2. Make the coordinate transform on the coordinates of points inside the resulting graphic.

I've wrapped these steps in a function:

ClearAll@implicitPlot
implicitPlot[eq_, range__, coordSys_, opt : OptionsPattern[ContourPlot]] := 
 Module[{coord}, 
  With[{plot = ContourPlot[eq, range, opt, PlotRange -> All], 
    trans = #[coord, #2] &[Function, 
       CoordinateTransform[coordSys -> "Cartesian", {coord[1], coord[2]}]] /. 
      coord[i_] :> Part[coord, i]}, 
   plot /. GraphicsComplex[coord_, rest_] :> 
     GraphicsComplex[trans[coord\[Transpose]]\[Transpose], rest]]]
    
implicitPlot[Cos@r == theta, {r, 0, 40}, {theta, -8 Pi, 8 Pi}, "Polar", 
 PlotPoints -> 100]

The advantage of this approach is, it allows us to directly set domain of definition under the interested coordinate system.

Another possible solution is to

1. ContourPlot the equation directly;

2. Make the coordinate transform on the coordinates of points inside the resulting graphic.

I've wrapped these steps in a function:

ClearAll@implicitPlot
implicitPlot[eq_, range__, coordSys_, opt : OptionsPattern[ContourPlot]] := 
 Module[{coord}, 
  With[{trans = #[coord, #2] &[Function, 
       CoordinateTransform[coordSys -> "Cartesian", {coord[1], coord[2]}]] /. 
      coord[i_] :> Part[coord, i]}, 
   Module[{plot = ContourPlot[eq, range, opt, PlotRange -> All]},
    plot[[1, 1]] = trans[plot[[1, 1]]\[Transpose]]\[Transpose]; plot]]]

implicitPlot[Cos@r == theta, {r, 0, 40}, {theta, -8 Pi, 8 Pi}, "Polar", 
 PlotPoints -> 100]

The advantage of this approach is, it allows us to directly set domain of definition under the interested coordinate system.

Another possible solution is to

1. ContourPlot the equation directly;

2. Make the coordinate transform on the coordinates of points inside the resulting graphic.

I've wrapped these steps in a function:

ClearAll@implicitPlot
implicitPlot[eq_, range__, coordSys_, opt : OptionsPattern[ContourPlot]] := 
 Module[{coord}, 
  With[{trans = #[coord, #2] &[Function, 
       CoordinateTransform[coordSys -> "Cartesian", {coord[1], coord[2]}]] /. 
      coord[i_] :> Part[coord, i]}, 
   Module[{plot = ContourPlot[eq, range, opt, PlotRange -> All]},
    plot /.  
     GraphicsComplex[coord_, rest_] :> 
      GraphicsComplex[trans[coord\[Transpose]]\[Transpose], rest]]]]

implicitPlot[Cos@r == theta, {r, 0, 40}, {theta, -8 Pi, 8 Pi}, "Polar", 
 PlotPoints -> 100]

The advantage of this approach is, it allows us to directly set domain of definition under the interested coordinate system.