1. Polar Coordinates

2. The Sign of r and a Point’s Location in Polar Coordinates.
The point P = (r, ) is located |r| units from the pole. If r > 0, the point lies on the terminal side of . If r < 0 the point lies along the ray opposite the terminal side of . If r = 0. the point lies at the pole, regardless of the value of .

3. Text Example
Plot the points with the following polar coordinates: a. (2, 135°)
Solution
a. To plot the point (r, ) = (2, 135°), begin with the 135° angle. Because 135° is a positive angle, draw  = 135° counterclockwise from the polar axis. Now consider r = 2. Because r > 0, plot the point by going out two units on the terminal side of . Figure (a) shows the point.
(2, 135°)
135º

5. Multiple Representation of Points
If n is any integer, the point (r, ) can be represented as
(r, ) = (r,  + 2n) or (r, ) = (-r,  +  + 2n )

6. Find another representation for in which:
r is positive and
r is negative and

7. Relations between Polar and Rectangular Coordinates
x = r cos 
y = r sin 
x2 + y2 = r2
tan  = y/x
P = (r, ) = (x, y)

8. Find the rectangular coordinates of the points with the following polar coordinates:b.

9. Converting a Point from Rectangular to Polar Coordinates (r > 0 and 0 <  < 2)
Plot the point (x, y).
Find r by computing the distance from the origin to (x, y).
Find  using tan = y/x with  lying in the same quadrant as (x, y).

10. Example
Find the polar coordinates of a point whose rectangular coordinates are (2, 4)
Solution:

11. Example
Find the polar coordinates of a point whose rectangular coordinates are (-1, √3)
Solution:

12. Example
Convert 2x-y=1 to a polar equation.
Solution:

13. Convert to Rectangular Form:

14. Convert to Rectangular Form:

15. Convert to Rectangular Form:

16. Convert to Rectangular Form: