1. Polar Coordinates 6.3

2. 6.3 Polar Coordinates Objectives
Plot points in the polar coordinate system
Find multiple sets of polar coordinate for a given point
Convert a point from polar to rectangular coordinates
Convert a point from rectangular to polar coordinates
Convert an equation from rectangular to polar coordinates
Convert an equation from polar to rectangular coordinates

3. 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 .

4. What is the relationship between cartesian coordinates & polar ones?
The radius = r is the hypotenuse of a rt. triangle that has base = x & height=y
Thus,
If x = horizontal leg & y = vertical leg of a right triangle, then

5. 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

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

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

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

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

11. 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).

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

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

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

15. Convert to Rectangular Form:

16. Convert to Rectangular Form:

17. Convert to Rectangular Form:

18. Convert to Rectangular Form: