Problem 35 - Essential Calculus

Question

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) over which the graph is traced only once. $$ r=3-4 \cos \theta $$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=3-4 \cos \theta\) is a circle, and the interval over which it is traced only once is \([0, \pi]\).

Step 1: Understand the Polar Equation

Given the polar equation \(r=3-4 \cos \theta\), \(\theta\) represents the angle measured from the positive x-axis and r is the distance from the origin. Identifying the form of the equation is the first step. In this case, the equation represents a circle.

Step 2: Draw the Graph using a Graphing Utility

Use a graphing utility to draw the polar graph according to the polar equation \(r=3-4\cos\theta\). The cosine function indicates a circle centered at 3 on the x-axis and with a radius of 4. The graph displays a closed circle, which represents that the graph will be traced only once over the specified interval of \(\theta\).

Step 3: Find the Interval for \(\theta\)

The graph of the function is traced once, so we need to find an interval where this happens. Generally, the cosine function has a period of \(2\pi\). However, due to the subtraction in the r equation, the period is traced twice within an interval of \(2\pi\). Hence, the interval where the graph of \(r=3-4\cos \theta\) is traced once is \([0, \pi]\).

Key Concepts

Cosine Function

The cosine function, represented as \(\cos \theta\), is one of the fundamental trigonometric functions. It describes the relationship between the angle \(\theta\) and the x-coordinate of a point on a unit circle. In polar coordinates, this function is essential because it helps determine the horizontal position of the point \((r, \theta)\).

When you see a polar equation like \(r = 3 - 4 \cos \theta\), the cosine function inside the equation affects how the graph will behave, particularly its symmetry and orientation. The graph of such a polar equation will be a circle, where the cosine function shifts and scales the graph. The '3' in the equation is the horizontal shift, while the '-4' scales the radius around the circle. The use of cosine implies that the graph will be symmetric about the horizontal axis, due to the properties of the cosine wave.

Understanding these properties of the cosine function is crucial, as it allows you to predict the graph's shape and behavior more effectively.

Graphing Utilities

Graphing utilities are tools that help you visually interpret mathematical equations, whether they are in Cartesian or polar form. When graphing polar equations, such as \(r = 3 - 4 \cos \theta\), a graphing utility can plot each point's position based on its polar coordinates \((r, \theta)\).

Features of graphing utilities include:

• Plotting graphs accurately and quickly.
• Switching between different coordinate systems (Cartesian and polar).
• Adjusting the intervals for \(\theta\) to see specific portions of the graph.
• Zooming in to analyze fine details of the plotted graph.

By using these features, you can better visualize the entire trace of the polar equation. For \(r = 3 - 4 \cos \theta\), the utility would display a circle, indicating how the cosine component influences the graph's geometry. Utilizing such technology allows for a deeper understanding and a practical approach to learning how equations translate into visual forms.

Trigonometric Intervals

In trigonometry, intervals are specific ranges of angles over which a function is evaluated or graphed. For polar plots like \(r = 3 - 4 \cos \theta\), understanding the intervals of \(\theta\) is key to knowing how the graph is traced.

The cosine function has a standard period of \(2\pi\), meaning it completes one full cycle over this interval. However, it's important to consider how the function is utilized in the given polar equation. The subtraction in the equation alters the behavior of the graph.

In the problem \(r = 3 - 4 \cos \theta\), the graph is fully traced once over an interval of \([0, \pi]\). This is because the effect of the cosine function within this range is enough to move through the necessary changes to make one complete cycle of the graph. If you extend the interval to \([0, 2\pi]\), you would see it traced twice.

Understanding intervals helps in correctly interpreting such graphs and avoiding mistakes when predicting the graph's path.