Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the curve represented by the parametric equations. Cycloid: \(x=\theta+\sin \theta, y=1-\cos \theta\)

Step-by-Step Solution

Answer

The parametric equations \(x=\theta+\sin \theta, y=1-\cos \theta\) represent a cycloid curve. After graphing the given parametric equations in a graphing utility with an appropriate \(\theta\) interval (suggested \([-2\pi, 2\pi]\)), we get the desired graph of the cycloid.

Step 1: Understand the Parametric Equations

In this scenario, the parametric equations are given as \(x = \theta + \sin \theta\) and \(y = 1 - \cos \theta\). Here, \(\theta\) represents the parameter, or 'input'. As \(\theta\) changes, the values of \(x\) and \(y\) will change resulting in a movement along the curve. These equations together describe the coordinates of the points on the cycloid curve.

Step 2: Use a Graphing Utility

A suggestion for a graphing utility to use would be 'Desmos' (a free online graphing calculator). In desmos, input the equations in the form 'parametric', 'x = \theta + \sin \theta' and 'y = 1 - \cos \theta'. For the interval of \(\theta\) you may want to use \([-2\pi, 2\pi]\) as a suggestion.

Step 3: Analyze the Result

After graphing, you will see a typical cycloid curve. The highest and lowest points occur at values of \(\theta\) that are multiples of \(2\pi\). The curve never goes below y=0 and for each full rotation of \(\theta\) (i.e., each increase of \(\theta\) by \(2\pi\)), the curve moves one unit to the right.