Problem 62 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the polar equation \(r=2+2 \cos \theta\) displays a circle with a radius of 2 centered at \(r=2\).

Step 1: understanding polar coordinates

Polar coordinates is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The polar coordinates are \(r\) and \(\theta\), where \(r\) refers to the distance from the reference point (the pole or origin) and \(\theta\) is the angle from the positive x-axis.

Step 2: Understand the given polar equation

In the polar equation, \(r=2+2 \cos \theta\), \(r\) changes with \(\theta\). The cos function has values in [-1,1], therefore, the minimum value of \(r\) is when \(\cos\theta = -1\) (i.e., \(r = 0\)) and the maximum value is when \(\cos\theta = 1\) (i.e., \(r = 4\)). As \(\theta\) varies from 0 to \(2\pi\), the graph will form a circle with a radius varying between 0 and 4.

Step 3: Graph the polar equation

Using a graphing tool, such as a graphing calculator or an online graphing utility, type in the equation \(r=2+2\cos\theta\). Set the mode to polar to ensure the graph is displayed correctly. As \(\theta\) varies from 0 to \(2\pi\), observe how \(r\) varies and thus how the graph is formed. The final graph will look like a circle centered at \(r=2\) with a radius of 2.