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Chapter 6: 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 based on these parametric equations will depict a hypocycloid path. Depending on the range of \(\theta\), one can notice the cyclical nature of the curve as its sweeps around to retrace the same path.

Step by step solution

01

Understand the Parametric Equations

The given parametric equations, \(x=3 \cos ^{3} \theta, y=3 \sin ^{3} \theta\) represent a function where \(\theta\) is the parameter. These equations describe all the points (x, y) on the curve by expressing x and y in terms of the variable \(\theta\).
02

Understand the Specific Function

In the given function, both x and y are expressed as cube powers of trigonometric functions. This implies that the function traces a hypocycloid path, which is a shape formed by a point on a small circle rolling within a larger one.
03

Using a Graphing Utility

Now, to put these equations in a graphing utility, input each equation separately into the y= function of the graphing calculator or software. Since these are parametric equations, make sure the calculator is set to 'parametric mode', if that setting exists. Enter the range for \(\theta\) which can vary from \(0\) to \(2\pi\) or further, depending on the specific curve and how much of it you want to display.

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