Polar and Cartesian coordinates relations
Conversion from polar to rectangular coordinates
Conversion from rectangular to polar coordinates
Equation of a circle in polar form
General equation of a circle in polar coordinates
Polar equation of a circle with a center on the polar axis running through the pole
Polar equation of a circle with a center at the pole
Polar coordinate system
The polar coordinate system is a two-dimensional coordinate system in which each point P on a plane is determined by the length of its position vector r and the angle q between it and the positive direction of the x-axis, where 0 < r < + oo  and  0 < q < 2p.
Polar and Cartesian coordinates relations,
Note, since the inverse tangent function (arctan or tan-1) returns values in the range  -p/2 < q < p/2, then
 for points lying in the 2nd or 3rd quadrant
 and for points lying in the 4th quadrant
Example:   Convert Cartesian coordinates (-1, -Ö3) to polar coordinates.
 Solution: and since the point lies in the 3rd quadrant, then

Equation of a circle in polar form
General equation of a circle in polar coordinates
The general equation of a circle with a center at
Using the law of cosine,
 r2 + r02 - 2rr0 cos(q - j) = R2
Polar equation of a circle with a center on the polar axis running through the pole
Polar equation of a circle with radius R and a center on the polar axis running through the pole O (origin).
 Since then,
 r = 2R cosq
Polar equation of a circle with a center at the pole
 Since,   r2 = x2 + y2   and   x2 + y2 = R2  then r = R
is polar equation of a circle with radius R and a center at the pole (origin).
Example:  Convert the polar equation of a circle  r = -4 cosq  into Cartesian coordinates.
Solution:       As,    r = -4 cosq
then    r2 = -4r cosq,
and by using polar to Cartesian conversion formulas,  r2 = x2 + y2   and   x = r cosq
 obtained is         x2 + y2 = -4x x2 + 4x + y2 = 0 or      (x + 2)2 + y2 = 4 the equation of a circle with radius R = 2 and the center (-2, 0).
Pre-calculus contents B