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Chapter 10: Problem 7

Explain three symmetries in polar graphs and how they are detected in equations.

Short Answer

Expert verified
In summary, there are three types of symmetry in polar graphs: 1. Symmetry about the polar axis (x-axis) - achieved by replacing 胃 with -胃 and checking if the equation remains unchanged. 2. Symmetry about the line 胃 = 蟺/2 (y-axis) - achieved by replacing 胃 with (蟺 - 胃) and checking if the equation remains unchanged. 3. Symmetry about the pole (origin) - achieved by simultaneously replacing r with -r and 胃 with (胃 + 蟺), and checking if the equation remains unchanged. Polar equations can exhibit one or more types of symmetry, which help simplify their analysis and graphing, and provide a deeper understanding of their properties.

Step by step solution

01

1. Symmetry about the polar axis (x-axis)

To check for symmetry about the polar axis in a polar equation, we should replace 胃 with -胃 and see if the equation remains unchanged. If the equation remains the same, the graph of the equation exhibits symmetry about the x-axis. For instance, let r(胃) = 3cos(胃) be a polar equation. To check for symmetry, we can replace 胃 with -胃: r(-胃) = 3cos(-胃) r(-胃) = 3cos(胃) (since cosine is an even function) Since the equation remains unchanged, 3cos(胃) is symmetric about the polar axis.
02

2. Symmetry about the line 胃 = 蟺/2 (y-axis)

To check for symmetry about the line 胃 = 蟺/2 (y-axis) in a polar equation, we need to replace 胃 with (蟺 - 胃) and see if the equation remains the same. If the equation remains the same, the graph of the equation displays symmetry about 胃 = 蟺/2. For example, let r(胃) = 2sin(胃) be a polar equation. To check for symmetry, we can replace 胃 with (蟺 - 胃): r(蟺 - 胃) = 2sin(蟺 - 胃) r(蟺 - 胃) = 2sin(胃) (since sine is an odd function) Since the equation remains unchanged, 2sin(胃) is symmetric about 胃 = 蟺/2.
03

3. Symmetry about the pole (origin)

To check for symmetry about the pole (origin) in a polar equation, we should replace r with -r and 胃 with (胃 + 蟺) simultaneously, and see if the equation stays the same. If the equation remains the same, the graph of the equation has symmetry about the pole. For instance, let r(胃) = 4sin(2胃) be a polar equation. To check for symmetry, we can replace r with -r and 胃 with (胃 + 蟺): r(胃 + 蟺) = 4sin(2(胃 + 蟺)) r(胃 + 蟺) = 4sin(2胃 + 2蟺) Now, applying the property of sine, sin(x + 2蟺) = sin(x), we find that: r(胃 + 蟺) = 4sin(2胃) Since replacing r with -r and 胃 with (胃 + 蟺) results in the same equation as the original, the polar equation 4sin(2胃) is symmetric about the pole.

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