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Chapter 6: Problem 10

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

Short Answer

Expert verified
The curve defined by \(r = 2 \cos 2 \theta\) is symmetric with respect to both the polar axis and the line \(\theta=\frac{\pi}{2}\), but not symmetric with respect to the pole.

Step by step solution

01

Test for Symmetry About the Polar Axis

Replace \(\theta\) with \(-\theta\) in the equation \(r=2 \cos 2 \theta\) and check if the equation remains the same.This gives \(r=2 \cos 2 (-\theta) = 2 \cos(-2 \theta)\). Because the cosine function is an even function (\(\cos(-x) = \cos(x)\)), the equation indeed remains the same, hence it is symmetric about the polar axis.
02

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

Replace \(\theta\) with \(\pi - \theta\) in the equation \( r=2 \cos 2 \theta\) and check for any changes.This gives \(r=2 \cos 2 (\pi - \theta)\). After expanding inside the cosine function we get \(r=2 \cos (2\pi - 2\theta)\), which can be simplified as \(r=2 \cos(-2\theta+2\pi)\). Because \(\cos(-2\theta+2\pi)=\cos(2\theta)\), the equation remains unchanged. Hence, the given curve is also symmetric about the line \(\theta=\frac{\pi}{2}\).
03

Test for Symmetry About the Pole

To test for symmetry about the pole, replace \(r\) with \(-r\) in the equation \(r=2 \cos 2 \theta\) and check if anything changes. Doing so gives \(-r=2 \cos 2 \theta\), which is not the same as the original equation. Therefore, the curve is not symmetric about the pole.

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