Problem 21 - 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. $$ x=4 \sec \theta, \quad y=3 \tan \theta $$

Step-by-Step Solution

Answer

The corresponding rectangular equation is \( x^2 (1 - (\frac{3y}{4})^2 ) = 16 \).

Step 1: Graph the Parametric Equations

With the provided parametric equations, plot the points by substituting a range of values for \( \theta \). The orientation of the curve is counterclockwise because as \( \theta \) increases, the point \( (x, y) \) moves counterclockwise.

Step 2: Identify the Relationship between x and y

Looking at the parametric equations, we see that \( x = 4 \sec \theta \) and \( y = 3 \tan \theta \). Expressing \( \sec \) and \( \tan \) in terms favorable for elimination leads to \( x = 4 / \cos \theta \) and \( y = 3 \sin \theta / \cos \theta \). Observe that there is \( \cos \theta \) in both expressions which will enable the possibility of dividing \( x \) by \( y \) or vice versa.

Step 3: Eliminate the Parameter

Divide the first equation by the second, and simplify. This yields: \( x/y = (4 / \cos \theta) / (3 \sin \theta / \cos \theta) \). Simplifying this yields \( x/y = 4/3 \sin \theta \). As we know that \( \sin^2 \theta + \cos^2 \theta = 1 \) , we can write it in terms of \( \cos \theta \) to get \( \cos^2 \theta = 1 - (\frac{3y}{4})^2 \) . Substituting this into \( x \), we obtain: \( x = 4 / \sqrt{1 - (\frac{3y}{4})^2} \). Squaring both sides yields \( x^2 = 16 / ( 1- (\frac{3y}{4})^2) \). This simplifies to the rectangular equation \( x^2 (1 - (\frac{3y}{4})^2 ) = 16 \).

Key Concepts

Eliminate the Parameter

Eliminating the parameter from parametric equations is a crucial step to connect these expressions to their rectangular form. In the given equations, we have:

• \( x = 4 \sec \theta \)
• \( y = 3 \tan \theta \)

To eliminate the parameter \(\theta\), we need to find a relationship between \(x\) and \(y\) that doesn't include \(\theta\). This process often involves trigonometric identities or algebraic manipulation.

Here, we can use the identity \(\sin^2 \theta + \cos^2 \theta = 1\). First, express the secant and tangent in terms of sine and cosine, like so:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Substitute back into the parametric equations. By manipulating and combining these, you express a relationship between \(x\) and \(y\) directly. This transforms them into a single rectangular equation, which describes the outcome without needing \(\theta\). This process not only simplifies the representation but also makes it easier to graphically understand the equation on the Cartesian plane.

Graphing Utility

Graphing utilities, such as graphing calculators or software tools, play a fundamental role in visualizing parametric equations. These tools help students understand how the equations behave and reveal the trajectory that the parameters describe.

For parametric equations like \( x=4 \sec \theta \) and \( y=3 \tan \theta \), you can input a range of \(\theta\) values into the graphing utility. These values typically range from where the functions are defined, e.g., considering the asymptotes that appear in the function. The graphing utility will then display how \(x\) and \(y\) pair for each \(\theta\), tracing the path on a graph.

Using such tools also allows students to observe the orientation or direction of the curve. Here, as \(\theta\) increases, the graph shows a counterclockwise motion. Thus, these utilities are invaluable for comprehending dynamic changes and spatial relationships within graphs.

Rectangular Equation

The rectangular equation is the equivalent representation of the parametric equations in standard Cartesian coordinates, composed of \(x\) and \(y\) without the parameter. It describes the same curve but in a more familiar \(x,y\)-coordinate form.

In this exercise, after eliminating the parameter \(\theta\) through manipulation and relevant trigonometric identities, you derive a rectangular equation. Beginning with:

• \( x = 4 / \cos \theta \)
• \( y = 3 \sin \theta / \cos \theta \)

We deduced \( x^2 (1 - (\frac{3y}{4})^2 ) = 16 \) from our earlier work.

This rectangular equation succinctly encapsulates the relationship between \(x\) and \(y\) without referencing \(\theta\). This conversion is key for analyses that prefer Cartesian planes, offering simplicity in graphing and interpreting the geometric nature of an equation.