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Chapter 6: Problem 9

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$r=4+3 \cos \theta$$

Short Answer

Expert verified
The polar function \(r = 4 + 3 \cos \theta\) is symmetric about the polar axis, but it's not symmetric about the line \(\theta=\frac{\pi}{2}\) and the pole.

Step by step solution

01

Testing Symmetry about the Polar Axis

Replace \(\theta\) with \(-\theta\) in the equation \(r = 4 + 3 \cos \theta\). If it becomes \(r = 4 + 3 \cos(-\theta)\), which is identical to the original equation since cosine function is even. Hence, the graph is symmetric about the polar axis.
02

Testing Symmetry about the Line \(\theta=\frac{\pi}{2}\)

Replace \(\theta\) with \(\pi - \theta\) in the equation \(r = 4 + 3 \cos \theta\). It becomes \(r = 4 + 3 \cos(\pi - \theta)\), which simplifies to \(r = 4 - 3 \cos(\theta)\). It is different from the original equation, therefore it is not symmetric about \(\theta=\frac{\pi}{2}\).
03

Testing Symmetry about the Pole

Replace \(r\) with \(-r\) in the equation \(r = 4 + 3 \cos \theta\). The equation becomes \(-r = 4 + 3 \cos(\theta)\), and it is not identical to the original equation. Hence, the graph is not symmetric about the pole.

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