Polar Coordinates and Graphs of Polar Equations

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). 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.

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

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Multiple Representations of Points There are many ways to represent the point additional ways to represent the point

Find the other representations for the point Copyright © by Houghton Mifflin Company, Inc. All rights reserved Stop

Warm Up. Graph and find the other 3 representations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved

7 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. Definitions of trigonometric functions

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Coordinate Conversion Coordinate Conversion (Pythagorean Identity) Example: Convert the point into rectangular coordinates (x, y).

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

Warm Up Convert the following point from polar to rectangular Convert the following point from rectangular to polar: (-4, 1) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

Convert rectangular to polar equations and polar to rectangular equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Graph polar equations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Converting Polar Equations to Rectangular 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

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Convert the polar equation into a rectangular equation.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Convert the rectangular equation x 2 + y 2 – 6x = 0 into a polar equation.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Graphs of Polar Equations Example: Graph the polar equation r = 2cos  –2 – r  The graph is a circle of radius 1 whose center is at point (x, y) = (1, 0).

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Definition: Symmetry of Polar Graphs If substitution leads to equivalent equations, the graph of a polar equation is symmetric with respect to one of the following. 1. The line 2. The polar axis 3. The pole Replace (r,  ) by (r,  –  ) or ( – r, –  ). Replace (r,  ) by (r, –  ) or ( – r,  –  ). Replace (r,  ) by (r,  +  ) or ( – r,  ). Example: In the graph r = 2cos , replace (r,  ) by (r, –  ). r = 2cos( –  ) = 2cos  The graph is symmetric with respect to the polar axis. cos( –  ) = cos 

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Example: Zeros and Maximum r- values Example: Find the zeros and the maximum value of r for the graph of r = 2cos  The maximum value of r is 2. It occurs when  = 0 and 2 . These are the zeros of r.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Special Polar Graphs: Limaçon Each polar graph below is called a Limaçon. –3 – –3 Note the symmetry of each graph. What does the symmetry have in common with the trig function?

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 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. And, again, note the symmetry. –5 5 3 –3 –5 5 3 –3 a a