Problem 55 - Precalculus

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=3(2-\sin \theta)$$

Step-by-Step Solution

Answer

The polar equation \(r=3(2-\sin \theta)\) creates a graph with a circular viewing window centered at the origin covering the radius from 3 units to 6 units, and the graph covers angles \( \theta \) from 0 to \(2\pi\) radians.

Step 1: Understanding Polar Equations

Polar equations are mathematical functions that describe curves in two-dimensional space using a radial coordinate and an angular coordinate: the distance from a reference point and the angle from a reference line. The polar equation of concern here is \(r=3(2-\sin \theta)\), where `r` is the distance from the origin and \( \theta \) is the angle.

Step 2: Graphing the Polar Equation

To graph the polar equation \(r=3(2-\sin \theta)\), you would typically use a graphing utility like a graphing calculator or online tool like Desmos or GeoGebra. Once you've inputted your equation into the utility, you should see a graph that represents the curve described by that equation.

Step 3: Describing the Viewing Window

The viewing window is the area of the coordinate plane that is visible in the graph. In the case of our polar equation, the viewing window usually has a radius of 3 units (because \(r=3(2-\sin \theta)\) swings between 3 and 6 units) and covers angles \( \theta \) from 0 to \(2\pi\) radians. It is the circular region centered at the origin with this radius.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical equations. When dealing with polar equations, especially like the one given here \( r=3(2-\sin \theta) \), graphical representation is crucial to better understand the curves they form. Most graphing utilities support polar coordinates, making it easier for students to experiment visually.

Some of the popular graphing utilities used online include:

• Desmos: An interactive web-based app excellent for plotting various types of graphs, including polar equations.
• GeoGebra: A versatile tool for both education and exploration in geometry and algebra that supports polar functions.
• Graphing Calculators: Physical devices like the TI-84 have built-in functionality to plot complex equations, including polar forms.

Inserting your equation into these graphing utilities helps you to instantly view its representation. This makes it much easier to analyze and interpret the behavior of polar curves—such as seeing where curves intersect or loop back on themselves.

Don't forget that graphing utilities also provide additional features, like adjusting the viewing window, zooming in or out, and even animating the plots to see dynamic changes as parameters are varied.

Viewing Window

The viewing window is crucial when graphing polar equations because it determines what part of the graph is visible. For the equation \( r=3(2-\sin \theta) \), setting your viewing window correctly ensures that you capture the entire curve.

Typically, a viewing window in polar coordinates is defined by:

• The range of angles \( \theta \), usually from 0 to \( 2\pi \) radians.
• The range of radii \( r \), which in this equation ranges from 3 to 6 units.

These settings imply that your graph will be captured in a circular area centered on the origin. Ensure that the window radius can accommodate the maximum value of \( r \) to avoid clipping parts of the graph.

Remember, altering the viewing window lets you focus in on areas of interest or take a step back for an overview. Experiment with different views to gain a deeper understanding and insight into the graph's behavior.

Trigonometric Functions

Trigonometric functions play an important role in polar equations, often determining the shape and nature of the curves. In the equation \( r=3(2-\sin \theta) \), the function \( \sin \theta \) influences the behavior of \( r \), creating a curve that varies with \( \theta \).

Key trigonometric functions include:

• Sine (\( \sin \theta \)): A function that determines vertical positions on a circle. It ranges from -1 to 1, affecting whether \( r \) expands or contracts.
• Cosine (\( \cos \theta \)): Often found in polar equations, it alters horizontal components.
• Tangent (\( \tan \theta \)): Though less common in polar coordinates, it influences angle determination and slope calculations.

In our polar equation, as \( \sin \theta \) fluctuates from -1 to 1, the value of \( r \) transitions from expanding to contracting, forming a unique shape in the graph. Understanding these trigonometric functions allows you to anticipate changes in the polar plot and interpret the significance of these changes in real-world scenarios.