Problem 51 - Calculus Volume 3

Question

Use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. $$ \begin{array}{l} x=\theta+\sin \theta \\ y=1-\cos \theta \end{array} $$

Step-by-Step Solution

Answer

The parametric equations represent a horizontal cycloid.

Step 1: Understand the Parametric Equations

The parametric equations are given as \( x = \theta + \sin \theta \) and \( y = 1 - \cos \theta \). These equations describe the movement of a point as \( \theta \) varies. Our task is to use these equations to graph the corresponding curve and identify the curve.

Step 2: Analyze the Graphing Process

To graph these equations using a graphing utility, input \( x = \theta + \sin \theta \) and \( y = 1 - \cos \theta \) into the software. Before graphing, set the range for \( \theta \). Commonly, \( \theta \) is ranged from 0 to \( 2\pi \) to capture the full cycle of sine and cosine functions.

Step 3: Plot the Curve

In the graphing utility, as \( \theta \) varies from 0 to \( 2\pi \), you'll notice the shape of the curve begins to form. Watch how the values of \( x \) and \( y \) change with respect to \( \theta \).

Step 4: Identify the Shape of the Curve

Upon graphing, the observed curve should resemble the shape of a "Horizontal Cycloid." Here, a cycloid can be identified as it has characteristic loops, created by the combination of \( \sin \theta \) and \( \cos \theta \) with \( \theta \). In this parametric form, each loop corresponds to one complete cycle around the circle described by \( \sin \theta \) and \( \cos \theta \).

Step 5: Confirm Identification

Finally, cross-check the observed graph with known cycloid shapes to confirm it matches a horizontal cycloid. A cycloid is typically the path traced by a point on the rim of a circular wheel as the wheel rolls along a straight line. The addition of \( \theta \) to \( \sin \theta \) and the contribution of \( \cos \theta \) create the distinctive shape.

Key Concepts

Cycloid

A cycloid is a fascinating curve that is created by tracking a point on the rim of a rolling circle along a straight path. Imagine a dot on a wheel and as the wheel rolls, this dot traces the cycloid path. In the context of parametric equations, such as those given by \( x = \theta + \sin \theta \) and \( y = 1 - \cos \theta \), the cycloid manifests from the interplay of these terms. These terms indicate the horizontal and vertical displacements of our point as the circle rolls.

• Horizontal displacement: \( \theta + \sin \theta \)
• Vertical displacement: \( 1 - \cos \theta \)

The phase difference between sine and cosine combined with \( \theta \) gives these loops. This undulating path is essential in multiple fields, from engineering to physics, where it can describe the shortest path between two points under certain conditions (brachistochrone problem).
Moreover, cycloids have unique properties, such as the tautochrone property, which states that objects dropped at different points along a cycloid will reach the lowest point simultaneously, regardless of their starting position.

Graphing Utilities

Graphing utilities are technological tools that help visualize complex mathematical equations, such as parametric equations. These tools are indispensable for analyzing the behavior of functions, within specified ranges, like \( \theta \) from 0 to \( 2\pi \). By utilizing features of graphing utilities, such as zoom and trace, users can delve deep into understanding the nuances of the equation plotted.

• Input capabilities for parameter setting
• Function visualization
• Zooming and scaling options
• Inspection and tracing of curve behavior

These utilities allow users to manipulate the viewing window, so the curve is observed more clearly. For students and educators, a graphing utility acts as a bridge between theoretical mathematics and its real-world representation. This visualization is especially helpful in identifying complex curves, including cycloids, offering insights into their formation and structure.

Trigonometric Functions

Trigonometric functions, such as \( \sin \theta \) and \( \cos \theta \), are fundamental in creating and analyzing parametric equations. In the context of our cycloid example, these functions dictate the oscillatory movements that develop the characteristic looping nature of the curve.

• \( \sin \theta \) is responsible for horizontal wave-like patterns.
• \( \cos \theta \) primarily contributes to the vertical component.

Each function has a period of \( 2\pi \), making them ideal for cyclic phenomena description. Since trigonometric functions span from -1 to 1, they inherently provide bounded periodic behavior, which is crucial for forming loops in cycloids. Understanding these functions is key in decomposing how these complex curves form and change over a given parameter range.

Curve Identification

Curve identification involves recognizing the unique features of a graph to determine what kinds of functions or equations create it. With parametric equations like those producing a cycloid, identification involves analyzing the presence of repeatable and recognizable patterns produced by trigonometric functions.

• Look for characteristic loops
• Recognize periodic repetition
• Check for corresponding features in known curves

The steps include graphing the equation, observing its shape, and comparing it to known geometric shapes. In this exercise, the plot of \( x = \theta + \sin \theta \) and \( y = 1 - \cos \theta \) is expected to reveal a series of arches typical of a cycloid. Recognizing these clues helps in understanding broader principles of mathematics and how distinct equations translate into observable curves on a graph.