Section 8.1 Polar Coordinates

1. Chapter 8 – Polar Coordinates and Parametric Equations Section 8.1 Polar Coordinates 8.1 - Polar Coordinates

2. Definitions • The polar coordinate system uses distances and directions to specify the location of a point in the plane. • We chose a fixed point O in the plane called the pole. • We then draw a ray from O called the polar axis. 8.1 - Polar Coordinates

3. Definitions • Each point P can be assigned polar coordinates P(r, ) where r is the distance from O to P.  is the angle between the polar axis and the segment . 8.1 - Polar Coordinates

4. Conventions • We use the following conventions:  is positive if measured in a counterclockwise direction from the polar axis.  is negative if measured in a clockwise direction from the polar axis. 8.1 - Polar Coordinates

5. Conventions • If r is negative, then P(r,) is defined to be the point that lies |r| units from the pole in the direction opposite to that given by  . 8.1 - Polar Coordinates

6. Examples – pg. 546 #’s 3 - 8 • Plot the point that has the given polar coordinates. 8.1 - Polar Coordinates

7. Examples – pg. 546 #’s 3 - 8 • Plot the point that has the given polar coordinates. 8.1 - Polar Coordinates

8. Examples – pg. 546 #’s 3 - 8 • Plot the point that has the given polar coordinates. 8.1 - Polar Coordinates

9. Multiple Representation of Points • The coordinates (r, ) and (-r,  + ) represent the same point. 8.1 - Polar Coordinates

10. Multiple Representation of Points • In fact each point has an infinitely number of representations in polar coordinates. • Any point P(r, ) can be represented by P(r,  + 2n) and P(-r,  + (2n + 1)) for any integer n. 8.1 - Polar Coordinates

11. Examples – pg. 546 #’s 9 – 12 • Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with r < 0 and the other with r > 0. 8.1 - Polar Coordinates

12. Examples – pg. 546 #’s 9 – 12 • Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with r < 0 and the other with r > 0. 8.1 - Polar Coordinates

13. Relationship Between Polar and Rectangular Coordinates • The connection between the two systems is shown below where the polar axis coincides with the positive x-axis. y P(r, ) P(x, y) y = r sin   x = r cos x 8.1 - Polar Coordinates

14. Relationship Between Polar and Rectangular Coordinates • To change from polar to rectangular coordinates, use the formulas: • To change from rectangular to polar coordinates, use the formulas 8.1 - Polar Coordinates

15. Example – pg. 546 # 31 • 31. Find the rectangular coordinates for the point whose polar coordinates are given. (5, 5) • 35. Convert the rectangular coordinates to polar coordinates with r > 0 and 0   < 2. (-1, 1) 8.1 - Polar Coordinates

16. Examples – pg. 546 • Find the rectangular coordinates for the point whose polar coordinates are given. 8.1 - Polar Coordinates

17. Examples – pg. 546 • Convert the rectangular coordinates to polar coordinates with r > 0 and 0   < 2. 8.1 - Polar Coordinates

18. Examples – pg. 546 • 44. Convert the equation to polar form. • 49. Convert the polar equation to rectangular coordinates. 8.1 - Polar Coordinates

19. Examples – pg. 546 • Convert the equation to polar form. 8.1 - Polar Coordinates

20. Examples – pg. 546 • Convert the polar equation to rectangular coordinates. 8.1 - Polar Coordinates