Problem 64 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the polar equation \(r=4+2 \cos \theta\) is a figure called a 'limaçon' which is symmetrical about the x-axis. At \(\theta=0\), r=6 and at \(\theta= \pi\), r=2.

Step 1: Understand the Polar Equation

In a polar equation, r is the distance from the origin (0,0) and \(\theta\) is the angle made with the positive x-axis. So, for our equation \(r=4+2 \cos \theta\), r varies as \(\theta\) changes.

Step 2: Use a Graphing Utility

To graph the polar equation, we need a graphing utility that supports polar coordinates. Set the mode to 'Polar' and input the equation as given, which is \(r=4+2 \cos \theta\). You will see a graph that is symmetrical about the x-axis.

Step 3: Analyze the Graph

Note the shape it makes. This polar equation gives a shape called a 'limaçon'. At \(\theta=0\), r=6 as cos(0) is 1 and at \(\theta= \pi\), r=2 as cos(\pi) is -1. The value of r keeps varying between these two distances from the origin as \(\theta\) changes from 0 to \(2\pi\).