Problem 71 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=\cos \frac{5}{2} \theta$$

Step-by-Step Solution

Answer

The detailed solution requires plotting the polar equation on a polar grid. As the equation doesn't easily convert to a Cartesian equation, tabulate \(\theta\) and \(r\) values to guide the plotting. The polar equation \(r = \cos \frac{5}{2}\theta\) creates a delicate flower-like structure with five petals, given the \(\frac{5}{2}\) multiplier.

Step 1: Understand Polar Coordinates

Polar coordinates are a type of coordinate system where each point in the space is defined by its distance \(r\) from the origin and its angle \(\theta\) from the positive horizontal axis. They are used when the relationship between two points is better expressed using their difference in angle and distance, instead of their difference in position within a Cartesian plane.

Step 2: Convert the Polar Equation Into Cartesian Form

To graph a polar equation, you can first convert it into its Cartesian form. This can be done by substituting \(r = \sqrt{x^2 + y^2}\), \(x = r\cos \theta\), and \(y = r\sin \theta\) into the equation. However, the equation \(r = \cos \frac{5}{2} \theta\) doesn't easily convert into a Cartesian equation, so pick several values for \(\theta\) and calculate corresponding \(r\) values instead.

Step 3: Create a Table of Values

Create a table with values of \(\theta\) and corresponding values of \(r\). This step requires you to input the \(\theta\) values into the given polar equation to get the \(r\) values. The table will guide you in plotting the graph of the polar equation.

Step 4: Plot the Points Using Polar Coordinates

Use the table of values (obtained in Step 3) to plot points onto the polar graph by marking a point for each (\(r, \theta\)) pair.

Step 5: Draw the Graph

After plotting all needed points, join them smoothly to draw the graph of the polar equation. The graph's shape will depend on your values in the table.