The PolarPlot
for your given function is actually
a = 2;
PolarPlot[Cos[x*a], {x, 0, Pi}]

The PolarPlot
that you show corresponds to
a = 4;
PolarPlot[Sin[x*a], {x, 0, 2 Pi}]

To display radius as a function of angle on a cartesian Plot
Plot[Abs[Sin[x*a]], {x, -Pi, Pi}]

Or as a LogPlot

The far-field radiation patterns that you show are not related to your function or your PolarPlot
EDIT: Using RegionPlot
RegionPlot[
-Sqrt[x^2 + y^2] <= Sin[a*ArcTan[y/x]] <=
Sqrt[x^2 + y^2],
{x, -1, 1}, {y, -1, 1},
MaxRecursion -> 5]

EDIT 2: Using DensityPlot
DensityPlot[Sin[a*ArcTan[y/x]],
{x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y, f},
-Sqrt[x^2 + y^2] >= f || f >= Sqrt[x^2 + y^2]],
PlotPoints -> 100,
MaxRecursion -> 3]
