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Chapter 7: Problem 16

Test for symmetry and then graph each polar equation. $$r=1+\sin \theta$$

Short Answer

Expert verified
The polar equation 'r = 1+ \sin \theta' is symmetrical with respect to the line \( \theta = \pi/2 \) and not symmetrical about the polar axis or the pole. Thus, the graph of this equation forms a cardioid shape.

Step by step solution

01

Symmetry with respect to the polar axis

Test for symmetry with respect to the polar axis, by replacing \( \theta \) with \( -\theta \) . If the equation remains the same, it's symmetrical along polar axis. Our equation becomes \( r = 1+ \sin (-\theta) = 1 - \sin \theta \). This isn't the same as our original equation, so it's not symmetrical with respect to the polar axis.
02

Symmetry with respect to the pole

Test for symmetry with respect to the pole, by replacing \( r \) with \( -r \) . If the equation remains valid, it's symmetrical. For our equation, \( -r = 1+ \sin \theta \) doesn't make sense as it implies \( r < 0 \) in the original equation, so it's not symmetrical with respect to the pole.
03

Symmetry with respect to the line theta = pi/2

Test for symmetry with respect to the line \( \theta = \pi/2 \) by replacing \( \theta \) with \( \pi - \theta \) . If the equation remains the same, it's symmetrical along that line. For our equation, this becomes \( r = 1+ \sin (\pi - \theta) = 1 + \sin \theta \), which equals to the original equation. Hence, it's symmetrical with respect to the line \( \theta = \pi/2 \).
04

Graph the equation

After determining the symmetry, you can plot the equation on a polar coordinate system. Notice that the radius \( r \) is a function of \( \theta \). For each value of \( \theta \), find the corresponding value of \( r = 1 + \sin \theta \) and plot this on the polar grid. The graph will reveal a cardioid shape as we move from \( \theta = 0 \) to \( \theta = 2\pi \).

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Most popular questions from this chapter

Explaining the Concepts Describe how to graph a polar equation.

Use a graphing utility to graph \(r=\sin n \theta\) for \(n=1,2,3,4,5\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of loops that occur corresponding to each value of \(n ?\) What is happening to the shape of the graphs as \(n\) increases? For each graph, what is the smallest interval for \(\theta\) so that the graph is traced only once?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There are no points on my graph of \(r^{2}=9 \cos 2 \theta\) for which \(\frac{\pi}{4}<\theta<\frac{3 \pi}{4}\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My work with complex numbers verified that the only possible cube root of 8 is 2
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