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Chapter 6: Problem 23

Test for symmetry and then graph each polar equation. $$r=2-3 \sin \theta$$

Short Answer

Expert verified
After testing for three types of symmetry, it was concluded that the polar equation \(r = 2-3\sin\theta\) does not exhibits any symmetry about the polar axis, symmetry about the pole, or symmetry about the line \(\theta = \pi/2\) . Consequently, each point on the graph must be plotted independently.

Step by step solution

01

Test for symmetry: Polar Axis

Substitute \(-\theta\) for \(\theta\) in the equation \(r =2 -3\sin\theta\) to obtain \(r' = 2 - 3\sin(-\theta) = 2 + 3\sin\theta\). Since \(r \neq r'\), the graph is not symmetrical about the polar axis.
02

Test for symmetry: Pole

Substitute \(-r'\) for \(r'\) in the equation \(r' = 2 + 3\sin\theta\) to get \(-r' = 2 + 3\sin\theta\), which simplifies to \(r' = -2 - 3\sin\theta\). However, this isn't the original equation so the graph is not symmetrical about the pole.
03

Test for symmetry: Line \(\theta = \pi/2\)

Replace \(\theta\) with \(\pi -\theta\) in \(r = 2 - 3\sin\theta\) to get \(r'' = 2 - 3\sin(\pi-\theta) = 2 + 3\sin\theta\). This transformation does not yield the original equation \(r = 2 - 3\sin\theta\), therefore, the graph isn't symmetrical about the line \(\theta = \pi/2\).
04

Graph the equation

To plot the graph for the polar equation \(r = 2 - 3\sin\theta\), plot the points for each value of \(\theta\) and connect the points. Since none of the symmetries have been identified for the equation, each point must be plotted independently to accurately represent the graph.

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