Problem 22 - College Trigonometry

Question

In Exercises \(21-24\), plot the set of parametric equations with the help of a graphing utility. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=4 \cos ^{3}(t) \\ y=4 \sin ^{3}(t) \end{array} \text { for } 0 \leq t \leq 2 \pi\right. $$

Step-by-Step Solution

Answer

Graph the parametric equations to see the astroid shape with counterclockwise orientation.

Step 1: Identify the Parametric Equations

The given parametric equations are:\[x = 4 \cos^3(t)\]\[y = 4 \sin^3(t)\]where \( t \) ranges from \( 0 \) to \( 2\pi \). These functions describe the coordinates \((x, y)\) of a point on the plane as \( t \) varies through this interval.

Step 2: Use a Graphing Utility

To graph these equations, use a graphing calculator or software capable of plotting parametric equations. Enter the equations for \( x \) and \( y \) to see how the points plot over the specified interval \( 0 \leq t \leq 2\pi \).

Step 3: Observe the Shape and Orientation

Upon graphing, you will notice that the curve follows the shape of an astroid. The orientation of the curve can be determined by observing that as \( t \) increases from \( 0 \) to \( 2\pi \), the curve moves in a counterclockwise direction.

Key Concepts

Graphing Utilities and Their Importance

Graphing utilities are powerful tools that help visualize mathematical functions, especially for complex forms like parametric equations. These tools can be physical graphing calculators or software applications such as GeoGebra or Desmos.

To use a graphing utility for parametric equations like in our exercise, you simply input the given equations for both x and y. In our case, you would input:

• \( x = 4 \cos^3(t) \)
• \( y = 4 \sin^3(t) \)

and set the parameter \( t \) to range from \( 0 \) to \( 2\pi \). This transforms the abstract concept of a parametric equation into a visual graph, making it easier to understand.

A graphing utility does more than just plot; it helps in checking the correctness of manual calculations and understanding how parameters affect the shape of the graph. Exploring with different parameter intervals can also reveal new insights about the behavior of equations.

Understanding Curve Orientation

In parametric equations, the orientation of the curve tells you the direction in which the curve is traced as the parameter \( t \) changes. This is an intuitive way to understand the flow or movement along the curve.

For the given parametric equations, as \( t \) increases from \( 0 \) to \( 2\pi \), you'll find that the astroid-shaped curve is traced in a counterclockwise direction. This means if you were traveling along the curve, starting from \( t = 0 \), you would move counterclockwise back to your starting point at \( t = 2\pi \).

Observing the curve's orientation is crucial in applications like physics or engineering, where understanding the path and its direction can indicate the trajectory of an object or flow of energy.

Astroid Shape - A Special Type of Curve

An astroid is a particular type of hypocycloid. It appears in many areas of mathematics and physics due to its unique properties and symmetry.

When graphed, our given parametric equations form an astroid. It's a star-shaped curve with four distinct 'cusp' points, where the curve sharply turns. The equations \( x = 4 \cos^3(t) \) and \( y = 4 \sin^3(t) \) control the shape and size of this astroid.

Each branch of the astroid is traced out as \( t \) ranges from one cusp back to itself. This shape can be found in various physical phenomena and engineering applications. The simplicity of plotting it using parametric equations makes it accessible for analysis and study in educational settings.

It's fascinating to compare an astroid with other shapes generated by similar parametric forms and observe the symmetry and repeating patterns that emerge across different curves.