Digital Lesson Polar Coordinates and Graphs of Polar

The polar coordinate system is formed by fixing a point, O, which is the

Plotting Points The point lies two units from the pole on the terminal side

There are many ways to represent the point additional ways to represent the 1

The relationship between rectangular and polar coordinates is as follows. y The point (x,

Coordinate Conversion (Pythagorean Identity) Example: Convert the point into rectangular coordinates. Copyright © by

Example: Convert the point (1, 1) into polar coordinates. Copyright © by Houghton Mifflin

Example: Convert the polar equation. into a rectangular Polar form Multiply each side by

Example: Graph the polar equation r = 2 cos . 0 r 2 1

If substitution leads to equivalent equations, the graph of a polar equation is symmetric

Example: Find the zeros and the maximum value of r for the graph of

Each polar graph below is called a Limaçon. 3 3 5 – 3 Copyright

Each polar graph below is called a Lemniscate. 3 3 – 5 5 –

Each polar graph below is called a Rose curve. 3 3 a 5 –

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Digital Lesson Polar Coordinates and Graphs of Polar Equations

The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). The polar axis is the ray constructed from O. Each point P in the plane can be assigned polar coordinates (r, ). e c n ta P = (r, ) is d d cte e O r ir d = = directed angle Pole (Origin) Polar axis 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. 2

The relationship between rectangular and polar coordinates is as follows. y The point (x, y) lies on a circle of radius r, therefore, r 2 = x 2 + y 2. (x, y) (r, ) r Definitions of trigonometric functions y Pole (Origin) x x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

Example: Convert the polar equation. into a rectangular Polar form Multiply each side by r. Substitute rectangular coordinates. Equation of a circle with center (0, 2) and radius of 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Example: Graph the polar equation r = 2 cos . 0 r 2 1 0 – 1 – 2 0 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 2 3 0 The graph is a circle of radius 2 whose center is at point (x, y) = (0, 1). 9

If substitution leads to equivalent equations, the graph of a polar equation is symmetric with respect to one of the following. 1. The line Replace (r, ) by (r, – ) or (–r, – ). 2. The polar axis Replace (r, ) by (r, – ) or (–r, – ). 3. The pole Replace (r, ) by (r, + ) or (–r, ). The graph is In the graph r = 2 cos , replace (r, ) by (r, – ). symmetric with respect to the r = 2 cos(– ) = 2 cos polar axis. cos(– ) = cos Example: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

Example: Find the zeros and the maximum value of r for the graph of r = 2 cos . 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. 1 2 3 0 11

Each polar graph below is called a Rose curve. 3 3 a 5 – 5 a – 3 The graph will have n petals if n is odd, and 2 n petals if n is even. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14