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Chapter 7: Problem 30

Test for symmetry and then graph each polar equation. $$r^{2}=9 \sin 2 \theta$$

Short Answer

Expert verified
The given polar equation \(r^{2}=9 \sin 2 \theta\) is symmetric about the x-axis only. For graphing, select some values of \(\theta\), compute respective \(r\) values and plot the points in the polar coordinate system.

Step by step solution

01

Test for Symmetry about the Pole

Substitute \(-\theta\) for \(\theta\), if the equation remains the same then it鈥檚 symmetric about the origin (pole). The equation becomes \(r^{2}=9 \sin 2 (-\theta)\) or \(r^{2}=9 \sin -2 \theta\) and since \(\sin -2 \theta\) is \( -\sin 2 \theta)\), the equation is not symmetric around the pole.
02

Test for Symmetry about the x-axis

Substitute \(-r\) for \(r\) and \(\pi-\theta\) for \(\theta\), if the equation remains the same, it is symmetrical about the x-axis. The equation becomes \((-r)^{2}=9 \sin 2 (\pi-\theta)\) or \(r^{2}=9 \sin (2\pi-2\theta)\) and is same as the original equation, indicating that the graph is symmetric about the x-axis.
03

Test for Symmetry about the y-axis

Substitute \(-r\) for \(r\) and \(-\theta\) for \(\theta\), if it reverts back to the original equation, it鈥檚 symmetric about the y-axis. The equation becomes \((-r)^2 = 9 \sin 2 (-\theta)\) or \(r^{2}=9 \sin -2 \theta\) which is not the same as the original equation, hence it is not symmetric around the y-axis.
04

Plot Critical Points

Once the symmetries are tested, start to graph the equation by selecting values for \(\theta\), computing corresponding \(r\) values and plot the points in a Polar Graph.

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