PPT Review 9.1-9.2

Polar Coordinates and Graphing

r = directed dista

nce

= directed angle Polar AxisO

,P r

Counterclockwise from polar axis to .OP

Plotting Points in the Polar Coordinate System

The point lies two units from the pole on the terminal side of the angle .

, 2,3

r

3

The point lies three units from the pole on the terminal side ofthe angle .

, 3,6

r

6

The point coincides withthe point .

11, 3,

6r

3,6

Multiple Representations of Points

In the polar coordinate system, each point does not have a unique

representation. In addition to 2 , we can use negative values

for . Because is a directed distance, the coordinates ,

and ,

r r r

r

represent the same point.

In general, the point r, can be represented as

r, , 2 or r, , 2 1

where is any integer.

r n r n

n

Example of Multiple Representations

from -360°≤ x ≤360°

1.) (100 , 50°) 2.) (1.2 , 70°)

(100 , -310°)(-100 , 230°)(-100 , -130°)

(1.2 , -290°)(-1.2 , 250°)(-1.2 , -110°)

Try plotting the following points in polar coordinates and find three additional polar representations of the point:

21. 4,

3

52. 5,

3

73. 3,

6

74. ,

8 6

Summary of Special Polar Graphs

1a

b

Limacon with inner loop

1a

b

Limacons:

Cardiod(heart-shaped)

DimpledLimacon

1 2a

b 2

a

b

Convex Limacon

cos

sin

0, 0

r a b

r a b

a b

Rose Curves:

cosr a b

petals if is odd

2 petals if is even

2

b b

b b

b

cosr a b sinr a b sinr a b

Circles and Lemniscates

cosr a sinr a 2 2 sin2r a 2 2 cos2r a

Circles Lemniscates

MATCH THE POLAR GRAPH WITH IT’S EQUATION

1. 2. 3. 4.

5. 6.

€

r = 3cosθ

€

r = 3

€

r = 2 + 2sinθ

€

r =1+ 2cosθ

€

r = sin3θ

€

r = sin4θ

5 3

2 4

6 1

MATCH THE POLAR GRAPH WITH IT’S EQUATION

1. 2. 3. 4.

€

r = sin θ 2( )

€

r = 1θ

€

r =θ sinθ

€

r =1+ 3cos(3θ )

2 4

31

Graphing a Polar Equation

6

3

2

2

3

5

6

7

6

3

2

11

6

2

2 3 2 3

circle

r 0 2 4 2 0 2 4 2 0

0

4sinr

Using Symmetry to SketchGraph: 3 2cos .r

6

3

2

2

3

5

6

3 3

3 3

Limacon DImpled

0 5

4

3

2

1

r

2r

Spiral of Archimedes

You can see that the graph is symmetric with respect to

the line .2

Graph This:

Rose Curve with 2b petals = 4 petals3cos2 .r

Graph each of these:

2 cosr 1 sinr Convex Limacon Cardioid Limacon

sin1r

€

0

€

1 − 0 =1

€

6

€

1

2

€

3

€

0.13

€

2

€

0

€

2π3

€

0.13

€

5π6

€

12

€

€

1

This type of graph is called a cardioid.

Scale of this graph is 0.25

0 5123

6

73.4

2

323

Let's let each unit be 1.

3

4

2

123

2

3023

3

2 22

123

6

527.1

2

323

1123 Let's plot the symmetric points

This type of graph is called a limacon without an inner loop.cos23r

0 3121

6

73.2

2

321

Let's let each unit be 1/2.

3

2

2

121

2

1021

3

2 02

121

6

573.0

2

321

1121 Let's plot the symmetric points

This type of graph is called a limacon with an inner loop.cos21r

0 212

6

1

2

12

Let's let each unit be 1/2.

4

002

3

1

2

12

2

212

Let's plot the symmetric points

This type of graph is called a rose with 4 petals. 2cos2r

0 004

6

32

2

34

Let's let each unit be 1/4.

4

414

3

32

2

34

2

004

This type of graph is called a lemniscate 2sin42 r r

0

9.1

2

9.1

0