Problem 65 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=4+2 \sin \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 4 + 2 \sin \theta\) represents a circle with the center at r=4 and radius of 2, offset by a sinusoidal variation due to the dependent \(\theta\) variable. The graph can best be obtained using a graphing utility which allows input of polar equations.

Step 1: Understand the Polar Equation

Analyze the equation \(r = 4 + 2 \sin \theta\), and understand that it's a circle centered at r=4 with radius 2. The variation of \(\theta\) causes points to move around the circle.

Step 2: Convert to Cartesian coordinates

If needed for understanding, the equation can temporarily be converted to cartesian coordinates using the transformation \(x = r \cos \theta\) and \(y = r \sin \theta\). Substituting \(r\) will then give a cartesians equation in x and y.

Step 3: Graph the Equation

Use a graphing tool to plot the equation. Start with \(\theta = 0\) and gradually increase the angle while calculating the corresponding r. The points thus obtained can be plot on the polar grid.

Step 4: Interpret the Graph

Verify the graph. For \(\theta = 0\) and \(\theta = \pi\), the radius r should be at maximal and minimal distance from origin respectively.