Question

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

Short 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-by-step solution

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.

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 \(\theta\).

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 \([0, \pi]\).