Problem 36 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=2+2 \cos \theta$$

Step-by-Step Solution

Answer

The graph of the given polar equation \(r=2+2\cos\theta\) is a limaçon with a loop which is on the positive side of the x-axis or polar axis.

Step 1: Understand the Graph Type

In polar coordinates, equation of the form \(r=a+bcos\theta\) or \(r=a+bsin\theta\) represents a Limaçon graph. For our equation \(r=2+2\cos\theta\), since the coefficient of cosine equals the constant term (i.e., both are 2), we are going to graph a limaçon with a loop.

Step 2: Identify the Key Points

To sketch the graph, first we need to identify the key points using the given values of \(\theta\). Consider the values \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\). Substituting these values into the given equation, we can get corresponding values of \(r\). This will give us points which we will use to sketch a representative plot.

Step 3: Sketch the Graph

Now we are to plot the graph. Plotting the points which we identified before, we will start at the pole corresponding to \(\theta = 0\) and proceed in the anticlockwise direction. The graph for our given equation \(r=2+2\cos\theta\) is a limaçon with a loop with its loop on the positive side of the x-axis or polar axis (since it is based on cosine function)

Key Concepts

Limaçon

A Limaçon is a unique type of curve that you might encounter when dealing with polar coordinates. These curves emerge from the equations of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Depending on the values of \( a \) and \( b \), Limaçons can appear quite differently. There's a fascinating aspect to note: if \( |a| < |b| \), the Limaçon will have an inner loop, as we see in the exercise equation \( r = 2 + 2 \cos \theta \), indicating that the loop exists when \(|a| = |b|\).
When dealing with Limaçons, consider the following types:

• Inner loop: Occurs when \( |a| < |b| \)
• Cardioid: Forms when \( |a| = |b| \)
• Dimpled: Appears when \( |a| > |b| \), but \( a eq b \)
• Convex: Seen when \( a > b \)

Understanding these types can help differentially sketch various Limaçon curves based on the relationship between \( a \) and \( b \).

Cosine Function

In the case of our Limaçon, the graph is described using the cosine function. When you have a cosine function in polar equations, the graph will be symmetric about the polar axis (equivalent to the positive x-axis). This implies symmetry about a line which is a significant feature when plotting the graph.
The cosine function contributes to the placement of the loop or dimple in the Limaçon when \( r = a + b \cos \theta \). This symmetry plays a pivotal role in the shape and orientation of the graph. For example:

• The presence of \( \cos \theta\) typically results in a horizontal symmetry, aligning the graph along the x-axis.
• The loop of the Limaçon for \( r = 2 + 2 \cos \theta \) lies along the positive x-axis.

Utilizing cosine ensures that calculations involving key angles (like \( 0, \pi, \frac{\pi}{2} \)) are straightforward, allowing easier identification of critical points to sketch the graph.

Graph Sketching

Sketching polar graphs, like the Limaçon, involves a few straightforward steps that ensure accuracy.

• Firstly, identify potential key angles such as \( 0, \frac{\pi}{2}, \pi, \) and \( 2\pi \).
• Substitute these values for \( \theta \) into the equation to calculate values for \( r \).
• Plot the points on polar coordinates, beginning with the smallest \( \theta \) and progressing forward.

Start your graph at \( \theta = 0 \); typically, you'll find the initial point on the positive x-axis. As you increase \( \theta \), plot each calculated point and join them smoothly, following the shape's defined curve.
Paying attention to symmetry helps in filling out the rest of the graph appropriately. Limaçons like ours with a loop require careful connecting of calculated points to display the loop when sketching, showing the full, elegant shape of the Limaçon.