Problem 22 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=1-2 \cos \theta$$

Step-by-Step Solution

Answer

The given polar equation is not symmetric about x-axis, y-axis or origin. The graph is a limaçon, a type of spiraling shape.

Step 1: Testing for symmetry

First, test for symmetry. To test for symmetry about the x-axis, replace \( \theta \) with \(- \theta \) in the given equation. If the equation remains the same, then it's symmetric about the x-axis. Similarly, for symmetry about the y-axis, replace \( \theta \) with \( \pi - \theta \). And for symmetry about the origin, replace \(\theta \) with \( \theta + \pi \). Testing the given equation doesn't yield the original equation in any of these tests so it is not symmetric about x-axis, y-axis or origin.

Step 2: Graphing the polar equation

To plot the graph, pick some representative values of \( \theta \) (like 0, \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\frac{3\pi}{4}\), \(\pi\)) and compute \(r\) value using the equation to get polar coordinates \((r, \theta)\). Then plot these coordinates on the polar graph. Make sure to plot a smooth curve going through all points. You may need to pick additional points if the graph is not straightforward.