copyright © cengage learning. all rights reserved. 9.6 graphs of polar equations
TRANSCRIPT
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Copyright © Cengage Learning. All rights reserved.
9.6 Graphs of Polar Equations
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What You Should Learn
• Graph polar equations by point plotting
• Use symmetry and zeros as sketching aids
• Recognize special polar graphs
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Introduction
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Example 1 – Graphing a Polar Equation by Point Plotting
Sketch the graph of the polar equation r = 4 sin by hand.
Solution:
The sine function is periodic, so you can get a full range of r-values by considering values of in the interval 0 2, as shown in the table.
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Example 1 – Solution
By plotting these points, as shown in Figure 9.70, it appears that the graph is a circle of radius 2 whose center is the point (x, y) = (0, 2).
Figure 9.70
cont’d
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Symmetry and Zeros
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Symmetry and Zeros
In Figure 9.70, note that as increases from 0 to 2 the graph is traced out twice. Moreover, note that the graph is symmetric with respect to the line = /2. Had you known about this symmetry and retracing ahead of time, you could have used fewer points.
Figure 9.70
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Symmetry and Zeros
The three important types of symmetry to consider in polar curve sketching are shown in Figure 9.71.
Figure 9.71
Symmetry with Respect Symmetry with Respect to the Polar Axis
Symmetry with Respectto the Pole
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Symmetry and Zeros
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Example 2 – Using Symmetry to Sketch a Polar Graph
Use symmetry to sketch the graph of r = 3 + 2 cos by hand.
Solution:Replacing (r, ) by (r, – ) produces
r = 3 + 2 cos (– )
= 3 + 2 cos
So, by using the even trigonometric identity, you can conclude that the curve is symmetric with respect to the polar axis.
cos(–u) = cos u
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Example 2 – Solution
Plotting the points in the table and using polar axissymmetry, you obtain the graph shown in Figure 9.72. This graph is called a limaçon.
Use a graphing utility to confirm this graph.
cont’d
Figure 9.72
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Symmetry and Zeros
The following are the quick tests for symmetry.
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Special Polar Graphs
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Special Polar Graphs
Several important types of graphs have equations that are simpler in polar form than in rectangular form.
For example, the circle r = 4 sin in Example 1 has the more complicated rectangularequation x2 + (y – 2)2 = 4.
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Special Polar Graphs
Several other types of graphs that have simple polar equations are shown below
Limaçons
r = a b cos , r = a b sin (a > 0, b > 0)
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Special Polar Graphs
Rose Curves
n petals when n is odd, 2n petals when n is even (n 2)
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Special Polar Graphs
Circles and Lemniscates
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Example 4 – Analyzing a Rose Curve
Analyze the graph of r = 3 cos 2.
Solution:
Type of curve: Rose curve with 2n = 4 petals
Symmetry: With respect to the polar axis, the line and the pole
Zeros of r : r = 0 when
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Example 4 – Solution
Using a graphing utility, enter the equation, as shown in Figure 9.75 (with 0 2).
You should obtain the graph shown in Figure 9.76.
Figure 9.76Figure 9.75
cont’d