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Chapter 6: Problem 39

Test for symmetry and then graph each polar equation. $$r=\frac{1}{1-\cos \theta}$$

Short Answer

Expert verified
The polar function \(r=\frac{1}{1-\cos \theta}\) has symmetry about the line theta = pi/2 and the graph of the function forms a lima莽on with a loop.

Step by step solution

01

Check Symmetry

First, check for symmetry. For symmetry with respect to the line theta equal to pi/2, replace theta with -theta in the equation. If the equation remains unchanged it has symmetry along this line. Here, we have: \(r=\frac{1}{1-\cos(-\theta)}\), which simplifies to \(r=\frac{1}{1-\cos \theta}\). The equation has not changed which demonstrates symmetry about the line theta = pi/2.
02

Graph the original equation

Next, graph the equation using various values for theta. The graph of \(r=\frac{1}{1-\cos \theta}\) is a lima莽on with a loop. Since there is an exclusion for \(r=1\), the graph does not extend beyond \(r=1\), this creates a dimple in the graph showing the edge of the loop. It may be useful to plot out points for different theta values to better visualize the exact loop shape. As this is a general method without the specified values, plot points between \(-\pi \leq \theta \leq \pi \) as your domain.
03

Label the Graph

Finally, after plotting the points, sketch the graph. Label the graph clearly with the equation and it's symmetry. Once the shape is drawn, the task is complete.

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