Problem 78 - Essential Calculus

Question

Use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Conchoid $$ r=2+\csc \theta $$ $$ y=1 $$

Step-by-Step Solution

Answer

The graph of the equation \(r = 2 + \csc(\theta)\) approaches but never touches the line y=1, proving that y=1 is an asymptote of the graph.

Step 1: Understand the polar function r = 2 + csc(θ)

Cosecant (\(\csc(\theta)\)) is the reciprocal of the sine function (\(\sin(\theta)\)). So, the given polar function can alternatively be written as \(r = 2 + \frac{1}{\sin(\theta)}\).

Step 2: Convert the polar equation to Cartesian coordinates

To convert the polar equation \(r = 2 + \frac{1}{\sin(\theta)}\) to Cartesian coordinates, use the conversion identities \(r^2 = x^2 + y^2\) and \(\sin(\theta) = \frac{y}{r}\). Substitute into the equation and simplify to get the Cartesian equation.

Step 3: Graph the Cartesian equation and y=1

Using a graphing utility, plot the Cartesian equation obtained in Step 2 and the line y=1. Notice how the curve approaches but never reaches the line y=1.

Step 4: Show that y=1 is an asymptote of the graph.

From the graph, it can be seen that the curve of the equation approaches but never reaches the line y=1, thus proving that y=1 is an asymptote of the graph.

Key Concepts

Polar to Cartesian Conversion

Converting polar equations to Cartesian coordinates involves using the transformation formulas that connect polar and Cartesian systems. Polar coordinates involve a radius, \(r\), and an angle, \(\theta\), measured from the positive x-axis, while Cartesian coordinates use an \(x\) and \(y\) coordinate.

• The conversion formulas are \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Also, we have the identity \(r^2 = x^2 + y^2\).
• Using these, polar equations like \(r = 2 + \csc \theta\) can be transformed into their Cartesian form.

To perform this conversion for the given polar equation, recognize that \(\csc(\theta)\) is the reciprocal of \(\sin(\theta)\). Therefore, the polar equation becomes \(r = 2 + \frac{1}{\sin(\theta)}\). Replacing \(\sin(\theta)\) with \(\frac{y}{r}\), we rewrite the equation as \(r = 2 + \frac{r}{y}\). Multiply through by \(y\) and rearrange to obtain the Cartesian form after simplification.

Graphing Utilities

Graphing utilities, such as graphing calculators and software, are powerful tools that aid in visualizing complex equations and their behaviors. When tasked with graphing equations like a conchoid, these utilities can quickly illustrate how curves interact with asymptotes.

• They allow students to input functions directly and view graphs instantaneously, showing behaviors such as asymptotes or intersections.
• Utilize features like zooming, tracing, and tangent line analysis to understand function behavior at points of interest.

Graphing \(y = 1\) alongside your converted Cartesian equation helps in spotting asymptotic behavior because it shows us how the graph approaches but never touches this line. This visual check confirms theoretical asymptotic analyses and helps students finding how these theoretical concepts manifest visually.

Cosecant Function

The cosecant function, denoted as \(\csc\theta\), is the reciprocal of the sine function: \(\csc\theta = \frac{1}{\sin\theta}\). Understanding the behavior and characteristics of \(\csc\theta\) is crucial as it often appears in polar equations.

• \(\csc\theta\) is undefined whenever \(\sin\theta = 0\), which happens at angles \(\theta = n\pi\), where \(n\) is an integer.
• These points of undefined behavior can cause asymptotes in graphs when involved in polar or Cartesian equations.

When the function \(r = 2 + \csc\theta\) is graphed, the inclusion of \(\csc\theta\) contributes to the asymptote at \(y = 1\). This is because as \(\theta\) approaches values where \(\sin\theta = 0\), \(\csc\theta\) approaches infinity, leading the curve to get closer to a line without ever touching it.