Problems were equations in rectangular form are converted to polar form, using the relationship between polar and rectangular coordinates, are presented along with detailed solutions.
Problem 1: Convert the equation
2x^{ 2} + 2y^{ 2}  x + y = 0
to polar form.
Solution to Problem 1:

Let us rewrite the equations as follows:
2 ( x^{ 2} + y^{ 2} )  x + y = 0

We now use the formulas giving the relationship between polar and rectangular coordinates: R^{ 2} = x^{ 2} + y^{ 2}, y = R sin t and x = R cos t:
2 ( R^{ 2} )  R cos t + R sin t = 0

Factor out R
R ( 2 R  cos t + sin t ) = 0

The above equation gives:
R = 0
or
2 R  cos t + sin t = 0

The equation R = 0 is the pole. But the pole is included in the graph of the second equation 2 R  cos t + sin t = 0 (check that for t = Pi / 4 , R = 0). We therefore can keep only the second equation.
2 R  cos t + sin t = 0
or
R = (1 / 2)(cos t  sin t)
Problem 2: Convert the equation
x + y = 0
to polar form.
Solution to Problem 2:

Use y = R sin t and x = R cos t into the given equation:
x + y = 0
R cos t + R sin t = 0

Factor out R
R ( cos t + sin t ) = 0

The above equation gives:
R = 0
or
cos t + sin t = 0

The equation R = 0 is the pole. But the pole is included in the graph of the second equation cos t + sin t = 0 since this equation is independent of R. We therefore keep only the second equation.
cos t + sin t = 0

The above equation may be written as.
tan t =  1

Solve for t to obtain
t = 3 pi / 4

All points of the form (R , 3 pi / 4) are on the graph of the above equation. It is the equation of a line in polar form.
More references on polar coordinates and trigonometry topics.
Polar Coordinates.
