Problem 18 - 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. $$ x=\cos \theta, y=2 \sin 2 \theta $$

Step-by-Step Solution

Answer

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

Step 1: Graph the Parametric Equations

Using a graphing utility, plug in the values of \( x=\cos \theta \), \( y=2 \sin 2 \theta \) and graph the curve. Remember to indicate the orientation of the curve: the curve starts where \( \theta = 0 \) and ends where \( \theta = 2\pi \)

Step 2: Substitute and Simplify

The goal is to eliminate the parameter \(\theta\). Notice that \( \sin^2 \theta = 1 - \cos^2 \theta \), which can be solved by substitution. As the equation defines \( x \) as \( \cos \theta \), substitute \( \cos \theta \) with \( x \) in the equation. Therefore, \( y = 2 \sin(2 \arccos x) \). Use the identity of \( \sin 2 \theta = 2 \sin \theta \cos \theta \). The equation becomes \( y = 4 \sin(\arccos x) \cos(\arccos x) \). Finally, substitute \( \sin(\arccos x) \) with \( \sqrt{1-x^2} \) and \( \cos(\arccos x) \) with \( x \). After re-arranging, the final answer is \( y = 4x\sqrt{1-x^2} \).

Step 3: Confirm the Conversion

After performing the conversion, it is always advisable to verify the equation. Re-plot the obtained rectangular equation on the same graphing software and verify it matches the plot created in step 1.

Key Concepts

Graphing Utility

When solving parametric equations like \( x=\cos \theta, y=2 \sin 2 \theta \) it's essential to visualize the curve to understand the relationship between the variables. A graphing utility, which can be a software tool or a graphing calculator, becomes invaluable here.

The utility allows you to plot the equations by inputting the parametric forms directly. It converts these inputs into a visual graph, showing the trajectory that the function traces. You usually start with an empty coordinate plane and as you enter the parametric equations, a curve is formed on the screen. One of the key features is being able to denote the orientation of the curve, which is the direction in which the curve is traversed as the parameter \(\theta\) increases. This is crucial for understanding the dynamics of the curve, especially when the parameter represents time or another sequential measurement.

Rectangular Equation

The term rectangular equation refers to the standard form of writing an equation with x and y variables, resembling the familiar y=mx+b or other non-parametric forms. Converting a parametric equation into a rectangular one makes it more straightforward to analyze and understand the behavior of the curve in relation to the Cartesian coordinate system.

To translate the example given, \(x=\cos \theta, y=2 \sin 2 \theta\), we want to represent \(y\) purely in terms of \(x\) without the parameter \(\theta\). This action simplifies the complexity behind the parametric representation and allows us to study the equation as a function \(y=f(x)\) in a more traditional sense.

Eliminate the Parameter

If your aim is to eliminate the parameter from parametric equations, it is akin to decoding a cryptogram; you're translating the language of the parametric equations into the more commonly used language of rectangular equations.

For example, taking our parametric equations \(x=\cos \theta\) and \(y=2 \sin 2 \theta\), we use trigonometric identities and inverse functions to replace \(\theta\) with expressions involving \(x\) and \(y\). By recognizing that \(y\) can be expressed in terms of \(x\), we manipulated the equations to arrive at the new form \(y = 4x\sqrt{1-x^2}\), successfully removing the parameter. This new equation describes the same curve, but now in terms of \(x\) and \(y\) solely, allowing it to be graphed as a standard function on a Cartesian plane without the need for the parameter \(\theta\).