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Chapter 6: Problem 24

Test for symmetry and then graph each polar equation. $$r=2+4 \sin \theta$$

Short Answer

Expert verified
After the symmetry checks, we conclude that the graph of the given polar equation \(r=2+4 \sin \theta\) has no symmetry, neither about the x-axis, y-axis, nor about the origin. The graph of the polar equation can be obtained by plotting different \(\theta\) values from \(0\) to \(2\pi\) and their corresponding \(r\) values.

Step by step solution

01

Symmetry Check - x-axis

Substitute \(-\theta\) for \(\theta\) in the original equation. If the resulting equation is identical to the original equation, then the graph is symmetric with respect to the x-axis. For this equation \(r=2+4 \sin \theta\), we replace \(\theta\) with \(-\theta\) to get \(r = 2+4 \sin(-\theta) = 2 - 4 \sin(\theta)\). Since this is not the same as the original equation, there's no x-axis symmetry.
02

Symmetry Check - y-axis

Replace \(\theta\) with \(-\theta\) and \(r\) with \(-r\) in the original equation. If the result is equivalent to the original equation, then the graph is symmetric with respect to the y-axis. For the provided equation \(r=2+4 \sin \theta\), we replace \(\theta\) with \(-\theta\) and \(r\) with \(-r\) to get \(-r = 2 - 4 \sin \theta\). This is also not the same as the original equation, so there isn't y-axis symmetry.
03

Symmetry Check - origin

Replace \(r\) with \(-r\). If the new equation can be rewritten to the original form, then the graph is symmetric with respect to the origin. For this equation \(r=2+4 \sin \theta\), replacing \(r\) by \(-r\) gives \(-r = 2+4 \sin \theta\). This is also not the original equation, indicating that the graph has no symmetry with respect to the origin.
04

Graphing

Since the graph does not have any symmetries, the equation can be plotted directly. The graph can be plotted by substituting different values of \(\theta\) in the interval from \(0\) to \(2\pi\), and then plotting the corresponding \(r\) values on polar coordinates.

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