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Chapter 10: Problem 53

Use a graphing utility to graph the curve represented by the parametric equations. Hypocycloid: \(x=3 \cos ^{3} \theta, y=3 \sin ^{3} \theta\)

Short Answer

Expert verified
The graph of the parametric equations x=3cos^3(胃), y=3sin^3(胃) will yield a hypocycloid.

Step by step solution

01

Recognizing the Parametric Equations

Recognize the given equations are parametric equations, which means each point in the plane is represented by a value of theta (胃) from a parameter domain, usually a segment (value range) in the real number line. Here, x is described by 3cos^3(胃) and y is described by 3sin^3(胃).
02

Set up the Graphing Utility

In your graphing utility, set up parametric graphing mode in order to graph curves represented by a pair of parametric equations. Usually, you can switch from standard graphing mode to parametric graphing mode in the settings of the tool you are using.
03

Input the Parametric Equations

Input the equations into the graphing utility. In the x= field input 3cos^3(胃) or 3cos^3(t) if your utility uses a different letter as parameter. Likewise, in the y= field input 3sin^3(胃) or 3sin^3(t).
04

Define the Range of 胃

Define the range of 胃 in your graphing utility. For most cases including this one, 胃 is set to range from 0 to 2蟺 to ensure complete plot of the curve.
05

Graph The Curve

Once the equations and range of 胃 are set properly, graph the curve. You should see a three-cusped figure, which is called a hypocycloid.

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