Problem 63 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=2+2 \sin \theta$$

Step-by-Step Solution

Answer

To graph the polar equation \(r = 2 + 2 \sin(\theta)\), one needs to create a table with different values for \(\theta\) and the corresponding 'r' values when plugged into the equation. These polar values are then plotted in a graph with appropriate scales. The resulting graph represents the behavior of the polar equation as \(\theta\) varies from 0 to \(2\pi\).

Step 1: Understand the Polar Equation

The given polar equation is \(r = 2 + 2 \sin(\theta)\). This equation describes a curve in polar coordinates, where r represents the distance from the origin (0,0) and \(\theta\) is the angle from the positive x-axis. 'r' changes with respect to \(\theta\), creating a unique graph.

Step 2: Create a Table for r and θ

Decide on the values of \(\theta\) you want to use. A common choice is to use increments such as 0, \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\frac{3\pi}{4}\), \(\pi\), \(\frac{5\pi}{4}\), \(\frac{3\pi}{2}\), \(\frac{7\pi}{4}\), and \(2\pi\). Now, plug each \(\theta\) value into our polar equation \(r = 2 + 2 \sin(\theta)\) to get the corresponding 'r' values.

Step 3: Plot the Polar Coordinates

For each (\(\theta\), r) pair in your table, plot a point that is 'r' units away from the origin, in the direction of \(\theta\). After plotting all the points, draw a smooth curve that goes through them. Since we're using a graphing utility, adjust the graph's window settings to ensure all points can be accurately viewed and compared.

Step 4: Analyze the Graph

The resulting graph represents the polar equation \(r = 2 + 2 \sin(\theta)\). Depending on the values computed for 'r', the graph will have a certain shape. Notice areas where the graph changes direction or shape as this is due to changes in the sign or value of 'r'. The graph is a unique representation of the polar equation.