Download - PPT Review 9.1-9.2
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PPT Review 9.1-9.2
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Polar Coordinates and Graphing
r = directed dista
nce
= directed angle Polar AxisO
,P r
Counterclockwise from polar axis to .OP
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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
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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
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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°)
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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
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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
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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
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Circles and Lemniscates
cosr a sinr a 2 2 sin2r a 2 2 cos2r a
Circles Lemniscates
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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
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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
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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
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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
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2r
Spiral of Archimedes
You can see that the graph is symmetric with respect to
the line .2
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Graph This:
Rose Curve with 2b petals = 4 petals3cos2 .r
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Graph each of these:
2 cosr 1 sinr Convex Limacon Cardioid Limacon
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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
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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
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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
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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
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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