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Chapter 6: Problem 43

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

Short Answer

Expert verified
The graph of \(r = 2 + 3\sin(2\theta)\) is not symmetric about the polar axis or about the polar origin. The graph must be plotted by generating and plotting points using the polar equation.

Step by step solution

01

Testing for Symmetry about Polar Axis

We test for symmetry about the polar axis by replacing \(\theta\) with \(-\theta\) in the equation. If the equation remains unchanged, then it is symmetric about the polar axis. In our equation, we get \( r = 2 + 3\sin(-2\theta) = 2 - 3\sin(2\theta) \), which is not the same as the original equation. Therefore, the graph is not symmetric about the polar axis.
02

Testing for Symmetry about the Polar Origin

To test for symmetry about the pole or origin, replace \(r\) with \(-r\). If we can manipulate it back to the original equation, then it is symmetric about the origin. Replacing \(r\) with \(-r\), we get \( -r = 2 + 3\sin(2\theta) \). Simplifying this equation does not yield the original equation, hence the graph is not symmetric about the origin.
03

Plotting the Polar Graph

Use the polar equation to generate points for the graph. Plot these points on polar coordinates, ensuring to correctly express the distance from the pole and the angle from the polar axis.

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