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Chapter 6: Problem 63

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

Short Answer

Expert verified
The graph of the polar equation \(r=2+2 \sin \theta\) forms a lima莽on with an inner loop. It reaches its maximum distance from the pole (4 units) when \(\sin \theta = 1\) and loops around the pole when \(\sin \theta = -1\), where it distance reduces to 0 units.

Step by step solution

01

Understand the Polar Coordinate System

In the polar coordinate system, a point in the plane is identified by its distance from the origin (r) and the angle it makes with the positive x-axis (胃) in the anticlockwise direction.
02

Identify the Equation Pattern

Equations of the form \(r = a + b \sin \theta\) or \(r = a + b \cos \theta\) create a type of graph known as a lima莽on. The key to identify here is that the plots will vary based on 卤a 卤 b. In our case we have +a + b configuration.
03

Visualize the Graph with a Utility

As this equation gives rise to a lima莽on, a graph representing a distorted circle is produced. With the given equation \(r = 2 + 2 \sin \theta\), graphing it on a polar system gives rise to a lima莽on with an inner loop. The maximum value of r is when \(\sin \theta = 1\), thus \(r = 2 + 2 = 4\), and the minimum value is when \(\sin \theta = -1\), thus \(r = 2 - 2 = 0\). The graph loops around the pole (origin) when r is 0.

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