Problem 60 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=4 \cos 6 \theta$$

Step-by-Step Solution

Answer

To graph the polar equation \(r=4 \cos 6 \theta\), understand that it describes a relationship where the radial distance \(r\) is dependent on the angle \(\theta\) and that this relationship is a cosine curve. Depending on the capabilities of your graphing utility, you may need to convert the equation to Cartesian form before graphing. The resulting graph should display a cyclic or repeating pattern due to the nature of the cosine function.

Step 1: Understand the equation and its form.

Polar equations are in the form \(r=f(\theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle formed with the positive x-axis. The given equation is \(r=4 \cos 6 \theta\), which implies that \(r\) (radial distance) is dependent on \(\theta\) (the angle). When \(\theta\) changes, the radius \(r\) changes according to the cosine function, which oscillates between -1 and 1.

Step 2: Convert the polar equation to Cartesian form.

This graphing utility might be more familiar with Cartesian equations. So let's convert our polar equation \(r=4 \cos 6 \theta\) to Cartesian form using the relations \(x = r \cos \theta\) and \(y=r \sin \theta\). However, this step is not necessary if your graphing utility supports Polar coordinates.

Step 3: Use the graphing utility.

Enter the equation into the graphing utility. If the utility supports polar coordinates, you can enter \(r=4 \cos 6 \theta\) directly. If it only supports Cartesian coordinates, input the equations derived from Step 2.

Step 4: Analyze the graph.

Once the graph is plotted, study it. The graph displays the relationship between the radial distance \(r\) and the angle \(\theta\). Because of the \(\cos\) function, the graph will show a cyclic or repeating pattern.