Problem 25 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the polar equation \( r=2 \cos 2\theta \) is a rose curve with four petals that are symmetric about the x-axis.

Step 1: Test for symmetry

First, let's test for symmetry. This step is crucial in graphing polar equations since it helps reduce the work and prevents potential mistakes. Symmetry about the x-axis, y-axis, and origin are the three types of symmetries that we should test for. If you replace \( \theta \) with \( -\theta \) in the equation, and if the original equation is unchanged (i.e., equivalent to the original equation), then the graph of the polar equation is symmetric about the x-axis. For \( r=2 \cos 2 \theta \), replacing \( \theta \) with \( -\theta \) yields \( r= 2\cos(-2\theta) \). Because the cosine function is even, \( \cos(-2\theta) = \cos2\theta \), which is equivalent to the original equation. Therefore, the polar equation is symmetric about the x-axis.

Step 2: Generate a table of points

Now that we know the polar equation is symmetric with respect to x-axis, we can think of graphing it. We start by generating a set of points. Given that the equation is of the form \( r=2 \cos (2\theta)\), we should plot points using values of \( \theta \) in the range \( [0, 2\pi) \). We only need to plot points between \( 0 \) and \( \pi \) due to the symmetry with respect to the x-axis. The other half of the curve will just be a reflection of the plotted points in this interval. Select several angles within this interval, calculate the corresponding values of \( r \) using the polar equation, and jot these pairs of values in a table.

Step 3: Plot the points

Once we've charted our points, it's time to place them on the polar grid. For each pair of \( (r, \theta) \) values in the table, \( r \) is the distance from the origin, while \( \theta \) is the angle this distance makes counterclockwise from the x-axis. Place all the plotted points on the grid.

Step 4: Sketch the graph

Once we've placed our points, we can sketch the graph using smooth curves that pass cleanly through the points. Due to the symmetry we identified in Step 1, reflect the first half of the curve over the x-axis to complete the graph.