Problem 53 - Calculus Early Transcendentals

Question

Show that the curve \(r=\sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x=1\) as a vertical asymptote. Show also that the curve lies entirely within the vertical strip \(0 \leqslant x<1\) Use these facts to help sketch the cissoid.

Step-by-Step Solution

Answer

The curve has a vertical asymptote at \(x=1\) and lies within \(0 \leq x < 1\).

Step 1: Convert to Cartesian form

To show the properties related to the x-axis, convert the polar equation \( r = \sin \theta \tan \theta \) to Cartesian coordinates. Use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). The equation becomes \( x = \sin \theta \tan \theta \cos \theta \). Simplify to \( x = \sin^2 \theta \), and \( y = \sin^3 \theta \).

Step 2: Analyze vertical asymptote

For a vertical asymptote, we check if \( x = 1 \) is approached as \( \theta \) increases. The equation \( x = \sin^2 \theta \) implies \( \sin \theta \rightarrow 1 \) as \( \theta \rightarrow \frac{\pi}{2} \). Thus, \( x \rightarrow 1 \) as an asymptote occurs at \( x = 1 \).

Step 3: Determine strip inclusion

To determine if the curve lies entirely within \( 0 \leq x < 1 \), note the relationship \( x = \sin^2 \theta \). The maximum value of \( \sin^2 \theta \) is 1, but \( x \) can never reach 1 as \( 0 \leq \sin^2 \theta < 1 \) for \( \theta \in (0,\pi) \). Therefore, the curve remains in the strip \( 0 \leq x < 1 \).

Step 4: Sketch the curve using asymptotic and boundary information

With the knowledge that \( x=1 \) is a vertical asymptote and the curve remains in \( 0 \leq x < 1 \), sketch the cissoid starting at the origin. As \( \theta \to \frac{\pi}{2} \), the curve approaches the asymptote without crossing it. The curve remains confined to the strip, oscillating between the boundaries as \( \theta \) varies.

Key Concepts

Vertical Asymptote

A vertical asymptote is a vertical line that a curve approaches but never touches or crosses. It represents a value of x where the function becomes unbounded. In the case of the Cissoid of Diocles, we identify the vertical asymptote by analyzing the behavior of the function at specific values of \( \theta \). As we convert the polar equation \( r = \sin \theta \tan \theta \) to its Cartesian form, we get \( x = \sin^2 \theta \). This expression suggests that as \( \theta \) approaches \( \frac{\pi}{2} \), \( \sin \theta \rightarrow 1 \) and consequently \( x \) tends towards 1. However, \( x \) can never actually reach 1. Thus, the line \( x = 1 \) acts as a vertical asymptote. The function approaches this line without ever crossing it.

Polar to Cartesian Conversion

Converting between polar and Cartesian coordinates is essential for analyzing curves like the Cissoid of Diocles. Polar coordinates describe a point's position based on the radius (r) from the origin and the angle (\( \theta \)) from the positive x-axis. On the other hand, Cartesian coordinates are expressed as (x, y).
To convert from polar to Cartesian, we use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This conversion reveals the relationship between the variables. For the given cissoid \( r = \sin \theta \tan \theta \), substitution leads to \( x = \sin^2 \theta \) and \( y = \sin^3 \theta \). These expressions help in studying the curve's properties and its behavior near vertical asymptotes.

Curve Sketching

Curve sketching involves plotting a graph of a function or relation based on its significant features. The Cissoid of Diocles is sketched by understanding its behaviors, such as approaching the vertical asymptote \( x=1 \) and being confined within a particular region.
Initially, we determine the boundary within which the curve lies. Here, analyzing \( x = \sin^2 \theta \) helps us find that \( 0 \leq x < 1 \). This indicates the curve never surpasses the line \( x = 1 \). To sketch the cissoid:

• Start from the origin when \( \theta = 0 \).
• As \( \theta \) increases, approach the vertical asymptote \( x=1 \).
• Ensure the sketch stays within the strip defined by \( 0 \le x < 1 \).

This method assists in visualizing how the curve behaves as \( \theta \) varies.

Trigonometric Functions

Trigonometric functions are fundamental in understanding curves expressed in polar coordinates. For the Cissoid of Diocles, trigonometric functions like sine (\( \sin \theta \)) and tangent (\( \tan \theta \)) are involved.
The equation \( r = \sin \theta \tan \theta \) uses these functions to define the curve's shape. Understanding how \( \sin \theta \) and \( \tan \theta \) behave is crucial. As \( \theta \) approaches \( \frac{\pi}{2} \), \( \sin \theta \) approaches 1, affecting the x-value in the Cartesian form. Knowing the properties of these trigonometric functions, such as their ranges and behavior at specific angles, aids in analyzing and sketching the curve.
This conceptual understanding ensures the effective conversion and analysis of curves from polar to Cartesian systems.