Problem 22 - Essential Calculus

Question

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=\cos ^{3} \theta, \quad y=\sin ^{3} \theta $$

Step-by-Step Solution

Answer

The graph is a loop with an anti-clockwise orientation. The rectangular equation for the given pair of parametric equations is \( (x+y)^3 = 1 \).

Step 1: Graphing the Parametric Equations

Use a graphing utility to graph the parametric equations. Here, the parameter is \( \theta \). For each value of \( \theta \), the point on the graph is (\( \cos ^{3} \theta \), \( \sin ^{3} \theta \)). This gives a general shape of the graph. For instance, for \( \theta = 0 \), point is (1, 0), for \( \theta = \frac{ \pi }{4} \), the point is ( \( \frac { \sqrt {2} }{2} \), \( \frac { \sqrt {2} }{2} \) ). Plot more points to get a clear picture of the graph.

Step 2: Indicating the Orientation of the Curve

The orientation of the curve begins from where \( \theta = 0 \) and follows the anticlockwise direction as \( \theta \) increases. For \( \theta = 2\pi \), we round back to the starting point, indicating the graph is a loop.

Step 3: Eliminating the Parameter

To eliminate the parameter (\( \theta \) ), make use of the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Cube both sides to get \( \sin^6 \theta + 3\sin^4 \theta \cos^2 \theta + 3\sin^2 \theta \cos^4 \theta + \cos^6 \theta = 1 \). Now replace \( \sin^3 \theta \) with y and \( \cos^3 \theta \) with x in the equation above and arrange terms to match with the identity \( a^3 + 3a^2b + 3ab^2 + b^3 = (a+b)^3 \). This gives us \( (x+y)^3 = 1^3 \). So the rectangular equation for the given pair of parametric equations is \( (x+y)=1 \).

Key Concepts

Graphing Utility

In the world of mathematics, a graphing utility is an incredibly useful tool, particularly when dealing with parametric equations. By graphing these equations, you can visually understand how a curve behaves over a certain range of the parameter. For this particular exercise, a graphing utility can be used to plot the curve defined by the parametric equations \(x = \cos^3 \theta \) and \( y = \sin^3 \theta \).
Using such a tool, you can input a range of values for \(\theta\) and observe the curve that forms.

• The graphing utility helps you see how these points connect smoothly to form a continuous curve.
• In the case of these equations, the graph forms a distinct loop-like shape.

This loop has a specific orientation: it starts at \((1,0)\) when \(\theta=0\) and follows an anti-clockwise path as \(\theta\) increases, eventually returning to the starting point by \(\theta=2\pi\).
This visual representation provided by graphing utilities is particularly helpful for understanding how the parameter \(\theta\) influences the coordinates of points on the graph. It also assists in determining the direction or orientation of the curve, which can be crucial for further analysis.

Rectangular Equation

When transitioning from parametric to rectangular equations, the goal is to eliminate the parameter and express the relationship between \(x\) and \(y\) directly. This offers a different perspective of the curve, focusing solely on the coordinates without considering the parameter.
In this exercise, the original equations are \(x = \cos^3 \theta\) and \(y = \sin^3 \theta\). By eliminating the parameter \(\theta\), we convert these into a rectangular equation.
To achieve this, we leverage the identity - \( \sin^2 \theta + \cos^2 \theta = 1 \). Cubing this identity inwardly, we can - express it as \( (x+y)^3 = 1^3 \).Ultimately, this simplification leads us to the rectangular equation \((x+y)=1\), which illustrates the direct connection between \(x\) and \(y\) on the curve, independent of \(\theta\).Understanding how to derive the rectangular equation from parametric equations is vital, especially in scenarios where the rectangular form is more convenient for further computations or analysis.

Pythagorean Identity

A fundamental concept in trigonometry is the Pythagorean identity, which is essential when dealing with trigonometric parametric equations. The identity is typically expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \).
This powerful identity helps us establish relationships between \(x\) and \(y\) in various mathematical contexts, such as solving the given parametric equations.
In this particular instance, the identity is instrumental in eliminating the parameter \(\theta\). By manipulating the basic identity: - Cubing the entire expression results in a relation that links higher powers of sine and cosine. - This manipulation allows us to substitute \(\sin^3 \theta\) with \(y\) and \(\cos^3 \theta\) with \(x\). This replacement leads us to a transformed equation, ultimately guiding us to the rectangular equation \((x+y)=1\).
Understanding and using the Pythagorean identity is crucial when working to eliminate parameters and simplify complex trigonometric expressions. Its versatility makes it a staple tool in mathematics, paving the way for a clearer understanding and analysis of parametric equations.