Problem 51 - Precalculus with Limits

Question

In Exercises 49-58, use a graphing utility to graph the polar equation. Describe your viewing window. \(r=\dfrac{5\pi}{8}\)

Step-by-Step Solution

Answer

The polar equation \(r = \dfrac{5\pi}{8}\) is a circle with radius approximately 1.96. Use a graphing utility to plot this circle. The viewing window should have a range of -2.5 to 2.5 both for x and y axes to ensure the entire circle fits neatly inside the window.

Step 1: Identify the type of the polar equation

The given polar equation \(r = \dfrac{5\pi}{8}\) is of the form \(r = k\), where k is a constant. This is the equation of a circle centered at the origin in polar coordinates with radius equal to k.

Step 2: Convert the constant in the equation

The given constant in the equation is in terms of \(\pi\), so convert it into a decimal form for ease of plotting. The constant \(k = \dfrac{5\pi}{8}\) is approximately 1.96.

Step 3: Graph the polar equation

To graph this equation using a graphing utility, plot a circle of radius approximately 1.96 centered at the origin. Remember to set the appropriate scale so that your circle is neatly visible within the viewing window.

Step 4: Describe the viewing window

The viewing window should be large enough to include the entire circle in the plot. Given the circle's radius is 1.96, for the horizontal (x) and vertical (y) view, a range of -2.5 to 2.5 should suffice. This will ensure the entire circle is represented clearly within the graph plane.

Key Concepts

Polar Coordinates

Polar coordinates offer a different system for plotting points and shapes in a plane. Unlike the rectangular coordinate system, which uses horizontal and vertical distances (x and y) to locate points, polar coordinates determine the location based on an angle and a distance from a central point. This system is especially useful in scenarios with circular symmetry, such as the orbits of planets or the rings of a tree trunk.

In polar coordinates, a point is described by a pair \( (r, \theta) \), where \( r \) is the radius, or the distance from the origin, and \( \theta \) is the angle, measured in radians or degrees, from the positive x-axis, going counterclockwise. To successfully navigate polar graphs, one must be familiar with angles and radii, and how these translate into the polar plane.

Circle in Polar Coordinates

A circle is a fundamental shape in geometry, and its representation in polar coordinates is elegantly simple. When graphing a circle centered at the origin in polar coordinates, the equation takes the form \( r = k \), where \( k \) is a constant that represents the radius of the circle.

The uniform radius means that no matter what the angle \( \theta \) might be, the distance from the origin stays constant, tracing out a perfect circle as \( \theta \) varies from 0 to \( 2\pi \) radians (or 0 to 360 degrees). This distinctiveness of polar coordinates allows students to appreciate how different coordinate systems can simplify certain equations and graphs.

Graphing Utility

Graphing utilities are sophisticated tools that aid students in visualizing complex algebraic and geometric concepts. These utilities take equations and render them visually as graphs, which can simplify understanding and interpreting the behavior of those equations. When dealing with polar coordinates, graphing utilities can instantly show the shapes corresponding to different polar equations, such as circles, spirals, and rose curves.

Utilizing these tools requires basic understanding of how to input polar equations and interpret the resultant graphs. Features like zoom, trace, and setting polar grids are invaluable for students exploring the polar coordinate system. Familiarity with these functions can make the study of polar equations more intuitive and rewarding.

Viewing Window Settings

The correct viewing window setting is crucial in graphing as it dictates what portion of the coordinate plane will be displayed. When setting up a viewing window for polar equations, ensure that the window is large enough to completely show the figure you're graphing but not so large that the figure becomes too small to analyze. For circles in polar coordinates, the viewing window should be slightly larger than twice the radius in all directions from the origin to comfortably contain the entire circle.

For the given problem, with a circle of radius roughly 1.96, setting the viewing range for both x and y to [-2.5, 2.5] will allow the whole circle to be visible. Be mindful that the viewing window can be adjusted to help analyze different parts of the graph in more detail, making it a versatile tool in your graphing utility.