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Chapter 6: Problem 54

Use a graphing utility to graph the curve represented by the parametric equations. Curtate cycloid: \(x=8 \theta-4 \sin \theta, y=8-4 \cos \theta\)

Short Answer

Expert verified
By graphing the parametric equations \(x=8 \theta-4 \sin \theta, y=8-4 \cos \theta\), the resulting plot is a curtate cycloid.

Step by step solution

01

Identifying the Parametric Equations

The parametric equations are \(x=8 \theta-4 \sin \theta\) and \(y=8-4 \cos \theta\). These equations give the x and y coordinates respectively in terms of \(\theta\).
02

Range of \(\theta\)

\(\theta\) is the parameter whose value varies as needed. Generally, in case of trigonometric functions \(\sin\) and \(\cos\), \(\theta\) is used to vary between \(0\) and \(2\pi\) to capture a complete cycle of the function.
03

Using a Graphing Utility

A graphing utility like a Graphing calculator or tool like desmos, geogebra etc., can be used to graph the parametric curve. Here, you need to input the equations into the graphing tool as specified, mentioning \(x\) as \(8 \theta-4 \sin \theta\) and \(y\) as \(8-4 \cos \theta\). Ensure to include the range for \( \theta \) as from \(0\) to \(2\pi\) to get the full graph of the given parametric equations.
04

Observing the Graph

After plotting the parametric equations, interpreting the graph is important. The drawn plot will demonstrate how the shape changes for both 'x' and 'y' as the parameter \( \theta \) varies. The graph in this case will be shaped as a curtate cycloid.

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Most popular questions from this chapter

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (4,3)\(;\) focus: (6,3)

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