polar coordinates and graphing r = directed distance = directed angle polar axis o counterclockwise...
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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
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
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 Sketch aPolar Graph
,r
,r
Symmetry with respect tothe line
.2
,r
,r
Symmetry with respect tothe Polar Axis.
Symmetry with respect tothe Pole
,r
,r
Tests for Symmetry
1. The line : Replace , with , - .2
2. The polar axis: Replace , with , .
3. The pole: Replace , with , .
r r
r r
r r
The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation.
Using Symmetry to SketchGraph: 3 2cos .r
Replacing , by , produces
3 2cos
3 2cos .
r r
r
6
3
2
2
3
5
6
3 3
3 3
Thus, the graph is symmetric with respect to the polar axis, and you need only plot points from 0 to .
Limaon
0 5
4
3
2
1
r
Symmetry Test FailsUnfortunately the tests for symmetry can guarantee symmetry, but there arepolar curves that fail the test, yet still display symmetry. Let’s look at the graphof Try the symmetry tests. What happens?2 .r
Original Replacement NewEquation Equation
2 r, with , 3 r r r
2 , with , 2r r r r
2 , with , 2r r r r
All of the tests indicate that no symmetry exists. Now, let’s look at the graph.
Not symmetric
about the line = .2
Not symmetric about the polar axis.
Not symmetric about the pole.
2r
Spiral of Archimedes
You can see that the graph is symmetric with respect to
the line .2
Quick Test for Symmetry
The graph of sin is symmetric with respect to the line .2
r a
The graph of cos is symmetric with respect to the polar axis.r a
Determine the effect of “a” on the graph of . sinr a
Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left.You may need to drag the graph slightly to the right.
Determine the effect of “a” on the graph of . cosr a
Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Determine the effect of “a” and “b” on the graph of sin .r a b
Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Determine the effect of “a” on the graph of Click on the graph below to open an interactive page.
Please note the interactive page may open with the graph to the far left.You may need to drag the graph slightly to the right.
sin .r a b
Determine the effect of “a” and “b’ on the graph of
Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
cos .r a b
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
Analyzing Polar Graphs
Analyze the basic features of
Type of Curve: Rose Curve with 2b petals = 4 petals
Symmetry: Polar axis, pole, and
Maximum Value of | r |: | r | = 3 when
Zeros of r: r = 0 when
3cos2 .r
.2
30, , ,
2 2
3, .
4 4
You can use this same process toAnalyze any polar graph.
Analyze and Sketch the graph:
2 cosr 1 sinr
Type of curve:
Symmetry:
Maximum | r |
Zeros of r
Convex Limacon Cardioid Limacon
sym. @ line =2
sym. @ polar axis
3,0 32,
2
none 0,2