Question

The graph of \(r=2 \cos (2 \theta) \sec (\theta) .\) is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

Short answer

The asymptotes of the strophoid are at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Step-by-step solution

Understanding the given polar equation

The polar equation given is \( r = 2 \cos(2\theta) \sec(\theta) \). This equation is in polar form, meaning that each point on the curve is determined by an angle \(\theta\) and radius \(r\). The equation involves trigonometric functions \(\cos\) and \(\sec\). Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so the equation simplifies to \( r = 2 \cos(2\theta) \cdot \frac{1}{\cos(\theta)} = 2 \cdot \frac{\cos(2\theta)}{\cos(\theta)} \).

Simplifying the expression

Simplify the expression further using the identity \( \cos(2 \theta) = 2 \cos^2(\theta) - 1 \), which yields \( r = 2 \left(2 \cos^2(\theta) - 1 \right) / \cos(\theta) \). Simplifying gives \( r = 4 \cos(\theta) - \frac{2}{\cos(\theta)} \).

Using a graphing utility to plot the function

To visualize this polar function and determine its asymptotes, use a graphing utility capable of handling polar plots. Input the simplified equation \( r = 4 \cos(\theta) - \frac{2}{\cos(\theta)} \) into the utility and plot it over a suitable range of \( \theta \), typically from \(0\) to \(2\pi\).

Analyzing the graph for asymptotes

Observe the graph produced by the utility. As \(\theta\) approaches values where \(\cos(\theta) = 0\), the term \(\frac{2}{\cos(\theta)}\) becomes undefined, suggesting vertical asymptotes may exist at \(\theta = \frac{\pi}{2} \) and \(\theta = \frac{3\pi}{2} \).

Identifying the asymptotes

The graph should reveal that the function approaches a line as \(\theta\) changes, indicating the location of the asymptotes. Based on the polar nature of the strophoid, expect vertical asymptotes at radial lines corresponding to \(\theta = \frac{\pi}{2} \) and \(\theta = \frac{3\pi}{2} \), which are confirmed visually by the graph.

Key concepts

Polar Graphs

Polar graphs represent equations and functions in polar coordinates, where points are defined by a distance from the origin (radius) and an angle from the positive x-axis. Unlike Cartesian graphs, which use x and y coordinates, polar graphs use a radius \( r \) and an angle \( \theta \). This system is particularly useful for plotting curves with circular or spiral shapes.

For example, the graph of the polar equation \( r = 2 \cos(2\theta) \sec(\theta) \) illustrates the strophoid, a distinct curve that is challenging to plot with Cartesian equations. Using polar coordinates allows for a more straightforward expression of these complex curves.

• \( r \) : distance from the origin.
• \( \theta \) : angle from the positive x-axis.
• Equations often involve trigonometric functions.

To plot these, graphing utilities capable of handling polar coordinates are often employed. They convert the polar equation into a visual curve, helping identify features like the shape and intersections.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles. They are key to simplifying and solving equations in polar coordinates. In our example, knowing the identity \( \cos(2\theta) = 2 \cos^2(\theta) - 1 \) allowed us to simplify \( r = 2 \cos(2\theta) \sec(\theta) \) by substituting \( \cos(2\theta) \).

Understanding identities like the above or \( \sec(\theta) = \frac{1}{\cos(\theta)} \) helps transform complex polar equations into simpler forms. This makes them easier to interpret and plot:

• \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• These identities help in handling divided terms or negative angles.

Simplifying the equation enables a clearer understanding of the graphical representation, particularly in identifying asymptotic behavior.

Asymptotes

Asymptotes are lines that a graph approaches but never truly reaches. In polar graphs, identifying asymptotes helps predict the behavior of the curve as \( \theta \) approaches certain values. In the case of the strophoid, the term \( \frac{2}{\cos(\theta)} \) suggests potential vertical asymptotes.

As \( \cos(\theta) \) goes to zero and \( \theta \) approaches \( \frac{\pi}{2} \) or \( \frac{3\pi}{2} \), this term becomes undefined. This implies vertical asymptotes at these angles because the radius \( r \) rapidly increases or decreases without bound. Visual graph analysis confirms these locations:

• Vertical asymptotes appear where \( \cos(\theta) = 0 \).
• On a polar graph, these correspond to \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).
• These lines represent boundaries the curve approaches but never intersects.

Recognizing asymptotes is crucial for understanding the full picture of a curve's structure and behavior.