1. Polar Coordinates and Graphs of Polar Equations

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The polar coordinate system is formed by fixing a point, O, which is the pole (or origin).  = directed angle Polar axis r = directed distance O Pole (Origin) The polar axis is the ray constructed from O. Each point P in the plane can be assigned polar coordinates (r,  ). P = (r,  ) r is the directed distance from O to P.  is the directed angle (counterclockwise) from the polar axis to OP.

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The point lies two units from the pole on the terminal side of the angle 123 0 3 units from the pole Plotting Points

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 There are many ways to represent the point 123 0 additional ways to represent the point

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 For each polar point, label it in two other ways:

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 (r,  ) (x, y)  Pole x y (Origin) y r x The relationship between rectangular and polar coordinates is as follows. The point (x, y) lies on a circle of radius r, therefore, r 2 = x 2 + y 2. Definitions of trigonometric functions

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Coordinate Conversion (Pythagorean Identity) Example: Convert the point into rectangular coordinates.

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Convert the following polar points to rectangular coordinates:

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Convert the point (1,1) into polar coordinates.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Convert the following rectangular points to polar coordinates:

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Convert the polar equation into a rectangular equation. Multiply each side by r. Substitute rectangular coordinates. Equation of a circle with center (0, 2) and radius of 2 Polar form

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Convert the following polar equations to rectangular equations.

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Convert the following rectangular equations to polar equations.

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Graph the polar equation r = 2cos . 123 0 2 0 –2 –1 0 1 20 r  The graph is a circle of radius 2 whose center is at point (x, y) = (0, 1).

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Graph the polar equation r = 3 cos .  r 0 30 60 90 120 150 180 210 240 270 300 330 360

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Graph the polar equation r = 3 + 2sin .  r 0 30 60 90 120 150 180 210 240 270 300 330 360

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Each polar graph below is called a Limaçon. –3 –5 5 3 5 3 –3

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Each polar graph below is called a Lemniscate. –55 3 –3 –5 5 3 –3

19. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Each polar graph below is called a Rose curve. The graph will have n petals if n is odd, and 2n petals if n is even. –5 5 3 –3 –5 5 3 –3 a a