Problem 254 - Calculus Volume 3

Question

Show that the curve \(r=\sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x=1\) as a vertical asymptote.

Step-by-Step Solution

Answer

The line \( x = 1 \) is a vertical asymptote of the curve.

Step 1: Convert to Rectangular Coordinates

First, let's convert the given polar equation to its Cartesian form. We have the polar equation as \( r = \sin \theta \tan \theta \). We know that \( x = r \cos \theta \) and \( y = r \sin \theta \). Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), replace \( \tan \theta \) in our initial equation: \( r = \sin \theta \cdot \frac{\sin \theta}{\cos \theta} \), so \( r = \frac{\sin^2 \theta}{\cos \theta} \). Therefore, in terms of \( x \) and \( y \), it translates to: \( y = x \tan \theta = r \sin^2 \theta = \frac{r \sin^2 \theta}{\cos \theta} \).

Step 2: Express Asymptote in Cartesians Terms

Now, we need to find if there is a vertical asymptote. A vertical asymptote in a Cartesian system occurs if as \( x \) approaches a certain value, the function becomes undefined or tends to infinity. Let's consider \( x = r \cos \theta = \frac{\sin^2 \theta}{\cos \theta} \cos \theta = \sin^2 \theta \). This implies that \( x = 1 - \cos^2 \theta \).

Step 3: Simplify to Find Asymptotic Behavior

Substitute \( \cos^2 \theta = 1 - x \) into the expression for \( x \). Since there's a singular point (where the expression is undefined) when \( \theta = \frac{\pi}{2} \) and \( \cos \theta = 0 \), we find that as \( \theta \) approaches \( \frac{\pi}{2} \), \( x = 1 \). Hence, \( x = 1 \) is a vertical asymptote of the curve.

Key Concepts

Polar to Cartesian conversion

Converting equations from polar coordinates to Cartesian coordinates can help us understand curves like the Cissoid of Diocles in more familiar terms.
Polar coordinates describe a point in the plane using a distance from the origin and an angle from the positive x-axis, while Cartesian coordinates use x and y values to describe a point.
In our problem, we begin with the polar equation \(r = \sin \theta \tan \theta\).
To convert this to Cartesian, recall the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). We also use \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This simplifies our polar equation to \(r = \frac{\sin^2 \theta}{\cos \theta}\).
The goal is to express both \(x\) and \(y\) using only these relationships.
Substituting \( x = \sin^2 \theta\) (from \( x = r \cos \theta = \sin^2 \theta \)), we express the curve purely in terms of \( x \) and \( y \) as \( y = \frac{r \sin^2 \theta}{\cos \theta} \), helping us visualize this curve in Cartesian form.

Vertical asymptotes

Understanding vertical asymptotes is key in analyzing the behavior of curves as they extend towards infinity or where they become undefined.
A vertical asymptote occurs at a particular \( x \) value when a function approaches infinity or becomes undefined as \( x \) approaches that value.
For the Cissoid of Diocles described by \( r = \sin \theta \tan \theta \), we aim to determine if there's an x value that causes the curve to exhibit this behavior.
As we convert the polar equation to \( x = \sin^2 \theta\), we see that we need to explore values for which \( x \) becomes undefined.
In particular, as \( \theta \to \frac{\pi}{2} \), we find \( \cos \theta = 0 \), leading to an undefined expression.
Therefore, the term \( x = 1 - \cos^2 \theta\) highlights that at the singularity \( x = 1 \), confirming that \( x = 1 \) is indeed a vertical asymptote. This asymptotic behavior is a significant feature of the curve.

Theta substitution

Theta substitution plays a crucial role when manipulating trigonometric expressions, helping to simplify the conversion between coordinate systems.
By using the angle \( \theta \) in substitutions, we leverage trigonometric identities to express complex polar equations in more manageable Cartesian forms.
In our task, theta substitution was used to account for \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), transitioning the equation into \(r = \frac{\sin^2 \theta}{\cos \theta} \).
Additionally, converting \(\sin^2 \theta\) and \(\cos^2 \theta\) using the Pythagorean identity assists in creating a bridge between polar and Cartesian systems.
Theta substitution simplifies understanding areas where asymptotic behavior such as singularities occurs.
As \(\theta\) approaches values like \(\frac{\pi}{2}\), the respective trigonometric identities provide clarity on why certain x-values cause undefined or infinite behavior, allowing us to identify vertical asymptotes.