1. 10.7 Polar Coordinates Adapted by JMerrill, 2011

2. Polar Coordinate Systems Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 You’ve all seen a polar coordinate system (the movies). Polar coordinates are used in navigation and look like

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). Definition: Polar Coordinate System  = 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.

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Plotting Points The point lies two units from the pole on the terminal side of the angle 123 0 3 units from the pole Plotting Points

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Multiple Representations of Points There are many ways to represent the point 123 0 We will only use 1 point instead of multiple representations.

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Polar and Rectangular Coordinate (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.

7. Coordinate Conversion To convert from polar to rectangular: x = r cosθ y = r sinθ To convert from rectangular to polar: tanθ = x 2 + y 2 = r 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

8. 8 Example: Coordinate Conversion Coordinate Conversion – the Relationship (Pythagorean Identity)

9. Example Convert to rectangular coordinates Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

10. 10 Example: Coordinate Conversion Example: Convert the point (1,1) into polar coordinates.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Graphs of Polar Equations 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). Radian mode Polar mode

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Special Polar Graphs: Limaçon Each polar graph below is called a Limaçon. –3 –5 5 3 5 3 –3

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Special Polar Graphs: Lemniscate Each polar graph below is called a Lemniscate. –55 3 –3 –5 5 3 –3

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Special Polar Graphs: Rose Curve 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