In the third section of Chapter 4 Goldstein discusses the use of polar coordinates and uses the right triangle definitions of the trigonometric functions to derive formulae for converting between polar coordinates and Cartesian coordinates. The angle q is indeterminate within an additive factor of an integer multiple of 360 degrees.
Some people also allow the use of negative values of r.
I distributed in class some sheets of polar graph paper. This is graph paper specially designed to facilitate making graphs of polar equations. An interesting feature of polar equations is the wide variety of very interesting shapes which appear. A common use of polar graphs occurs in atomic physics or chemistry when the polar or spherical coordinate representations of wave functions are plotted. The points are then plotted on polar graph paper.