1. Demana, Waits, Foley, Kennedy
6.4
Polar Coordinates

2. What you’ll learn about
Polar Coordinate System
Coordinate Conversion
Equation Conversion
Finding Distance Using Polar Coordinates
… and why
Use of polar coordinates sometimes simplifies
complicated rectangular equations and they are useful in
calculus.

3. The Polar Coordinate System
A polar coordinate system is a plane with a point O, the pole, and a ray from O, the polar axis, as shown. Each point P in the plane is assigned polar coordinates as follows: r is the directed distance from O to P, and  is the directed angle whose initial side is on the polar axis and whose terminal side is on the line OP.

4. Example: Plotting Points in the Polar Coordinate System

5. Solution

6. Finding Polar Coordinates
Starting at the pole, give four different polar coordinates that would require you to turn 600 (or π/3 radians) and travel a distance of 10 units.

7. Finding Polar coordinates

8. Finding all Polar Coordinates of a Point

9. Coordinate Conversion Equations

10. Example: Converting from Polar to Rectangular Coordinates

11. Solution

12. Example: Converting from Rectangular to Polar Coordinates

13. Solution

14. Example A: Rectangular to Polar

15. Example A: Rectangular to Polar

16. Example: Converting from Polar Form to Rectangular Form

17. Solution

18. Polar to rectangular
Express r = 5 sec in rectangular coordinates and graph the equation

19. Express r = 5 sec in rectangular coordinates and graph the equation
Solution
r = 5 secθ
r = 5(1/cosθ)
r cos θ = 5
x = 5

20. r = 2sinθ r2 = 2rsinθ (mult each side by r to get r2 rsinθ)
r = 2sinθ
r2 = 2rsinθ (mult each side by r to get r2 rsinθ)
x2 + y2 = 2y
x2 + y2 - 2y = 0 (complete the square)
x2 + (y - 1)2 = 1
This is a circle centered at (0, 1) with a radius of 1

21. Solve and graph: r = 2 + 2cosθ
r2 = 2r + 2rcosθ (Substitute)
x2 + y2 = 2r + 2x
x2 + y2 -2x = 2r (square each side to get r2)
(x2 + y2 -2x)2 = 4r2
(x2 + y2 -2x)2 = 4(x2 + y2) (STOP HERE)
x4 - 4x3 + 2x2y2 +4x2 - 4xy2 + y4 = 4x2 + 4y2
x4 - 4x3 + 2x2y2 - 4xy2 - 4y2+ y4 = 0

22. Example: Converting from Polar Form to Rectangular Form

23. Solution

24. Solution Continued