Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=\frac{6}{2 \sin \theta-3 \cos \theta}$$

Step-by-Step Solution

Answer

Without specific software to graph this polar equation, a complete representation of the viewing window cannot be given. However, typically for a polar function, the range of \( \theta \) from 0 to \(2\pi\) is sufficient. The range of r values will vary depending upon the shape of the graph.

Step 1: Determine the Range of \( \theta \)

Polar coordinates are typically represented as (r, \(\theta\)), where r is the radius and \(\theta\) is the angle in radians. For most polar graphs, we consider \(\theta\) to range from 0 to \(2\pi\). Therefore, the range of \(\theta\) for this problem will span from 0 to \(2\pi\).

Step 2: Enter the Equation into the Graphing Utility

Input the given equation \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\) into the graphing tool correctly. Remember to set the tool to Polar mode because we are graphing a polar equation. Most graphing tools have this functionality. The way to set it varies based on the tool being used.

Step 3: Graph and Analyze the Polar Equation

Use the graphing tool to sketch the polar plot. Look for the features of the plot to understand the shape. Normally, the graphing tool will provide the plot based on the range of \( \theta \) provided.

Step 4: Describe the Viewing Window

The viewing window in this case refers to the range of values for both r and \(\theta\) that allow for the entire graph to be seen. This will depend on the specific behavior and features of the graph. Carefully adjust the viewing window to ensure all parts of the graph are visible. Describe the values of \( \( \theta \) and r that allow for the entire graph to be seen.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle, which is a different approach from the Cartesian coordinate system that employs x and y axes. In polar coordinates, a point is described as \(r, \theta\), where \(r\) is the distance from the origin, or the pole, and \(\theta\) is the angle measured in radians from the polar axis (analogous to the x-axis in Cartesian coordinates).

Unlike Cartesian coordinates, which represent points with perpendicular reference lines, polar coordinates reflect the circular nature of the system. They are commonly used in scenarios where the relationship between two points is cleaner or more intuitive when expressed as angles and distances.

Radians

Radians are a unit of angular measurement used in polar coordinates. One radian is the angle created when the radius of a circle is wrapped around its circumference. It is another way to measure angles, as opposed to degrees. There are \(2\pi\) radians in a full circle, which is equivalent to 360 degrees.

When graphing polar equations, the angle, \(\theta\), is often measured in radians. This sine and cosine functions in trigonometry take angles in radians as arguments, and as such, graphing utilities require input in radians. Different mathematical concepts and functions behave in a more predictable and simpler manner when using radians instead of degrees.

Graphing Utility

A graphing utility is an invaluable tool in visualizing mathematical concepts, particularly when dealing with complex functions and equations like those in polar form. When entering a polar equation into a graphing utility, it is essential to switch the mode to 'Polar' to ensure the graph is interpreted and displayed correctly.

The utility plots points following the polar coordinate system, automatically computing and displaying the graph according to the range of \(\theta\) provided. It’s crucial to familiarize oneself with the specifics of their graphing utility, as the input method for equations can differ between programs or calculators.

Viewing Window

The viewing window of a graphing utility defines the portion of the coordinate plane that will be displayed. Setting an appropriate viewing window is critical in analyzing the behavior of the graph accurately. If the window is too narrow, important features of the graph might be missed. On the other hand, if the window is too broad, the graph may appear flattened and details can become indistinguishable.

To properly set the viewing window for polar graphs, adjust the range of \(\theta\) to cover at least one full period of the graph, typically from 0 to \(2\pi\) radians. Then, adjust the radial distance, \(r\), to ensure the entire graph fits within the viewing area. It may take several adjustments to capture the entirety and the nuances of the graph's features.