Problem 19 - 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. $$ \begin{array}{l} x=4+2 \cos \theta \\ y=-1+\sin \theta \end{array} $$

Step-by-Step Solution

Answer

The graph of the parametric equations follows a counterclockwise rotation. The corresponding rectangular equation is \((x-4)^2/4\) + \((y+1)^2 = 1\).

Step 1: Graph the Parametric Equations

Feed the equations into the graphing utility. For this, the inputs for x and y will be as follows: \(x=4+2 \cos \theta \) and \(y=-1+\sin \theta\). The plot of these equations will give us the look and the orientation of the curve.

Step 2: Note the Orientation of the Curve

The curve will start from the point where \(\theta = 0\) and will continue in the direction as \(\theta\) increases. Notice that as \(\theta\) ranges from \(0\) to \(2\pi\), the curve completes one full rotation counterclockwise.

Step 3: Eliminate the Parameter

To write a rectangular equation, we need to eliminate the parameter \(\theta\). As we know, \(\cos^2 \theta\) + \(\sin^2 \theta\) = 1. Therefore, the equation becomes \((x-4)^2/4\) + \((y+1)^2 = 1\).

Key Concepts

Rectangular Equations

Parametric equations involve a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. In our exercise, the parametric equations are given as \(x = 4 + 2\cos \theta\) and \(y = -1 + \sin \theta\). These describe a curve in terms of the parameter \(\theta\).

To convert these parametric equations into a rectangular equation, which is more traditional, we need to eliminate the parameter \(\theta\). A rectangular equation is typically written in terms of \(x\) and \(y\) only.

• Start with known trigonometric identities: \( \cos^2 \theta + \sin^2 \theta = 1\).
• Use the given equations to express \(\cos \theta\) and \(\sin \theta\) in terms of \(x\) and \(y\).
• Substitute into the identity to eliminate \(\theta\), achieving our rectangular equation: \(\frac{(x-4)^2}{4} + (y+1)^2 = 1\).

This is now a more familiar ellipse equation.

Graphing Utility

Graphing utilities, such as graphing calculators or software tools, are invaluable for visualizing mathematical equations and data. They allow users to input parametric equations, like those given in the problem, and visually see what the curve will look like.

With our parametric equations \(x = 4 + 2\cos \theta\) and \(y = -1 + \sin \theta\), a graphing utility can help plot these over a specific range of \(\theta\), typically from 0 to \(2\pi\). This range shows one full cycle of the curve.

• Input the parametric equations into the utility.
• Adjust the range for \(\theta\) to see the full curve.
• Observe the graph to understand the behavior and shape of the curve.

This visual representation makes it easier to explore complex mathematical concepts and aids in identifying any properties or features of the curve.

Curve Orientation

Orientation is an important aspect when dealing with parametric curves. It tells us the direction in which the curve is traced as the parameter increases. In our case, as \(\theta\) increases from 0 to \(2\pi\), the curve described by the parametric equations \(x = 4 + 2\cos \theta\) and \(y = -1 + \sin \theta\) is traced out.

For this particular example:

• Start with \(\theta = 0\), which corresponds to the initial point on the graph.
• As \(\theta\) increases, observe the direction of movement along the curve.
• In this exercise, the curve is traced counterclockwise.

Understanding orientation helps in knowing how the curve progresses and is useful in distinguishing between similar-looking curves.

Parameter Elimination

Eliminating the parameter from parametric equations is a key process to convert them into a standard form like the rectangular equation. This involves removing \(\theta\) to express the relationship solely between \(x\) and \(y\).

Here’s how elimination is done with our equations:

• Recognize the trigonometric identity: \( \cos^2 \theta + \sin^2 \theta = 1 \).
• Use the equations \(x = 4 + 2\cos \theta\) and \(y = -1 + \sin \theta\) to isolate \(\cos \theta\) and \(\sin \theta\).
• Substitute these into the trigonometric identity to eliminate \(\theta\), resulting in \(\frac{(x-4)^2}{4} + (y+1)^2 = 1\).

This helps in deriving the familiar form of the equation, allowing easier analysis and graphing without needing the parameter.