Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=8 \sin \theta \cos ^{2} \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=8 \sin(\theta) \cos^{2}(\theta)\) displays a polar rose pattern. The best viewing window for this polar equation is likely to be [-10,10] for the x-axis and [-10,10] for the y-axis.

Step 1: Input the Polar Equation

To plot the graph, open your chosen graphing utility and select the option to plot polar equations. Input the polar equation \(r = 8 \sin(\theta) \cos^{2}(\theta)\).

Step 2: Define Theta Range

Set the range of \(\theta\) to display for the full range of the graph, typically from \(-\pi\) to \(+\pi\).

Step 3: Adjust Viewing Window

After the equation is plotted, adjust the viewing window to clearly see the details of the plot. This may require zooming in or out and shifting the view.

Step 4: Describe Viewing Window

After setting the viewing window, describe it. Pay attention to the minimum and maximum for x-axis and y-axis that clearly display the graph of the polar equation.