Polar Coordinate System
      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
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, -3) 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,  
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