Problem 56 - Precalculus with Limits

Question

In Exercises 49-58, use a graphing utility to graph the polar equation. Describe your viewing window. \(r=2\ \cos(3\theta-2)\)

Step-by-Step Solution

Answer

The short answer to this question is subjective as it depends on the specific graphing utility used. However, an example response could be: the viewing window for the graph of \(r=2\ \cos(3\theta-2)\) falls within the range of \(\theta\) values from 0 to \(2\pi\), and the \(r\) values vary depending on the observed results after graphing.

Step 1: Recall the Polar Coordinates

Polar coordinates are a type of coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. In this case, \(r=2\ \cos(3\theta-2)\) describes a polar equation, where \(r\) is the distance from the pole and \(\theta\) is the angle from the polar axis.

Step 2: Input the equation into the graphing utility

The equation \(r=2\ \cos(3\theta-2)\) must be input into the graphing utility, allowing us to see its shape.

Step 3: Set the viewing window

Select an appropriate viewing window that captures the essential features of the graph. A standard window for polar graphs is \(\[\theta\] \in [0, 2\pi]\) (a full circle) and \(r\) being dependent on the equation and results seen on the graph.

Step 4: Describe the viewing window

After observing the graph, describe the viewing window giving the range of \(\theta\) and \(r\) that you used. This will help in identifying the critical points and complete shape of the graph.

Key Concepts

Polar Coordinates

When we come across curves and lines in a planar geometry, we often use Cartesian coordinates—the familiar x and y—to describe their positions and shapes. However, in mathematics, we sometimes switch to a different system known as polar coordinates, especially when dealing with circular or spiral patterns like those found in certain natural phenomena or engineering applications.

Picture yourself standing at the origin of a graph, the 'pole', with a line extending rightwards from you representing 0 degrees. This line is the 'polar axis'. Now, any point on this plane can be described by how far away and in what direction it is from you. The distance from the pole is denoted as 'r' (radius), and the direction is given by an angle 'theta' (θ), measured in radians or degrees, from the polar axis.

In the polar equation given, \(r=2\cos(3\theta-2)\), 'r' changes with θ in a way that creates a unique path when plotted. This is significant in complex designs like gear teeth patterns or in understanding wave behaviors.

Graphing Utility

To visualize what the polar equation \(r=2\cos(3\theta-2)\) actually represents, we turn to a graphing utility, which is a software tool that helps students and mathematicians alike to plot and study the behavior of various equations. These utilities often come equipped with features that allow for the input of equations in many forms, including polar.

Once the polar equation is input into the graphing utility, it computes the values of 'r' for various values of theta and plots these points accordingly. This process transforms the abstract mathematical equation into a concrete visual representation. The key benefit of using a graphing utility lies in its ability to handle complex equations that are often difficult to draw by hand, revealing intricate patterns and relationships through the graphical output.

Viewing Window

The viewing window in a graphing utility defines the segment of the coordinate plane that you'll be observing. Think of it like adjusting the zoom and pan of a camera; you want to capture the subject—in this case, the graph—in as much detail as necessary without losing context.

For polar equations, establishing a proper viewing window is crucial because different segments of the graph can exhibit vastly different behaviors. A standard window for polar plots usually includes \(\theta\) from 0 to \(2\pi\) to encompass a full rotation around the pole, but the range for 'r' can widely vary depending on the given equation. For the equation \(r=2\cos(3\theta-2)\), we would set the window to ensure that the maximum and minimum values of 'r' are visible. This may require some trial and error or prior knowledge of the potential shape of the graph. It's like framing the perfect shot: once you find the right window, the full beauty of the equation's graph is revealed.