Problem 60 - Precalculus

Question

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. $$r=5+4 \cos \theta$$

Step-by-Step Solution

Answer

The interval for θ in which the graph of the polar equation \(r =5 + 4 \cos θ\) is traced only once is \(0 ≤ θ < 2π\).

Step 1: Understand the polar function

The given polar equation is \(r = 5 + 4 \cos \theta\). This is a limacon (a type of polar graph) with a loop, because cos(θ) varies between 1 and -1, so r varies between 1 and 9. This will affect the range of θ later on.

Step 2: Graph the equation

Use a graphing utility to graph the equation. Begin by plotting points at various θ values, such as \(θ = 0, \frac{π}{4}, \frac{π}{2}, \frac{3π}{4}, π, \frac{5π}{4}, \frac{3π}{2}, \frac{7π}{4}\), and \(2π\). You can then connect these points to form a complete graph.

Step 3: Identify the interval for θ

Now, identify the interval for θ for which the graph is traced only once. By observation, you can see that the graph is traced exactly once over the interval from \([0, 2π)\), that is, \(\theta\) must be within the range \(0 ≤ θ < 2π\)

Key Concepts

Limacon

A limacon is a type of polar curve and is represented by equations of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). It's a fascinating shape because it can take on different forms, depending on the values of \(a\) and \(b\). Here's a rundown:

• A limacon can appear with a loop, a heart shape known as a cardioid, or even look like a dimpled circle.
• When \( |b| > |a| \), the limacon will have an inner loop, just like in your exercise, where \( r = 5 + 4 \cos \theta \) results in a limacon with a loop.
• When \( |b| = |a| \), it forms a cardioid, which is like a heart shape without a loop.
• When \( |b| < |a| \), the limacon appears dimpled or as a gently curved shape.

The key takeaway here is to observe the coefficients \(a\) and \(b\) to predict the limacon's form. Ultimately, becoming comfortable with these variations will simplify interpreting and graphing polar equations.

Graphing Utility

Graphing utilities are essential tools in visualizing equations, especially in complex and intricate versions like polar forms. A graphing calculator or software can:

• Provide an immediate picture of the equation's graphical features by plotting points on a coordinate plane.
• Allow for experimentation with different angles and coordinate systems, offering a deeper understanding of polar graphs.
• Facilitate a step-by-step exploration of how angular changes in \( \theta \) impact the radius \( r \), resulting in a complete graph.

For the equation \( r = 5 + 4 \cos \theta \), a graphing utility aids in transferring theoretical knowledge into a visual format. By entering the equation, users can observe the loop of the limacon, demonstrating its periodic behavior without manually plotting each point. This turns into a powerful learning moment, reinforcing the interplay between algebraic expressions and their geometric counterparts.

Interval for Theta

Finding the right interval for \( \theta \) in polar equations ensures that the graph is drawn precisely once, avoiding repetition. In general, polar graphs can be traced more than once over typical ranges unless calculated specifically. For the limacon in the given problem, \( r = 5 + 4 \cos \theta \), the task is to identify where this tracing happens only once. Here are the steps one might take:

• Observe the polar curve with values from \( \theta = 0 \) to \( \theta = 2\pi \), noting how \( r \) varies with \( \cos \theta \).
• Recognize that since the cosine function completes one full cycle in \( [0, 2\pi) \), the limacon also completes its loop within this interval.
• Conclude that \( [0, 2\pi) \) is the minimum interval over which the graph is traced a single time, capturing all symmetries the equation exhibits.

This method ensures a comprehensive understanding of repeating patterns and behavior unique to polar equations, allowing for accurate graphing and analysis.