Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 (θ) from a parameter domain, usually a segment (value range) in the real number line. Here, x is described by 3cos^3(θ) and y is described by 3sin^3(θ).

Step 2: 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.

Step 3: Input the Parametric Equations

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

Step 4: Define the Range of θ

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

Step 5: Graph The Curve

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