Polar and Cartesian coordinates relations
Conversion from polar to rectangular coordinates
Conversion from rectangular to polar coordinates
Polar coordinates of a point
Equation of a line in polar form
Lines parallel to the axes, horizontal and vertical lines
Lines running through the origin or pole (radial lines)
Polar equation of a line
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 line in polar form
Lines parallel to the axes, horizontal and vertical lines
 Lines parallel to the y-axis A vertical line, x = c is represented by the equation r cosq = c   or
 Lines parallel to the x-axis A horizontal line, y = c is represented by the equation r sinq = c   or
Lines running through the origin or pole (radial lines)
The equation of a line through the origin or pole that makes an angle a with the positive x-axis
 is represented by the equation q = a in polar coordinates.
Polar equation of a line
 As
the polar equation of a line
where p is the distance of the line from the pole O and j is the angle that the segment p makes with the polar axis.
Example:  Write polar equation of the line passing through points (-4, 0) and (0, 4).
Solution:   Using polar equation of a line
Proof, use of Cartesian to polar conversion formulas.
 The intercept form of the line or
-x + y = 4,       y = x + 4
 -r cosq  + r sinq  = 4,     r (sinq  - cosq )  = 4,
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).
College algebra contents A