Problem 21 - Thomas Calculus

Question

Graph the limacons. Limacon ("lee-ma-sahn") is Old French for "snail." You will understand the name when you graph the limacons in Exercise 21. Equations for limacons have the form \(r=a \pm b \cos \theta\) or \(r=a \pm b \sin \theta .\) There are four basic shapes. Limacons with an inner loop a. \(r=\frac{1}{2}+\cos \theta \quad\) b. \(r=\frac{1}{2}+\sin \theta\)

Step-by-Step Solution

Answer

Both limacons have inner loops due to \( b > a \).

Step 1: Identify the Equation Type

The given equation for part a is \( r = \frac{1}{2} + \cos \theta \), which follows the general form \( r = a + b \cos \theta \) with \( a = \frac{1}{2} \) and \( b = 1 \). Similarly, for part b, \( r = \frac{1}{2} + \sin \theta \) follows the form \( r = a + b \sin \theta \) with \( a = \frac{1}{2} \) and \( b = 1 \). In both these cases, \( b > a \), indicating the limacon will have an inner loop.

Step 2: Sketch Part a: Limacon with Cosine

Graph the equation \( r = \frac{1}{2} + \cos \theta \) by plotting coordinates. Start by setting specific values for \( \theta \), such as \( 0, \pi/2, \pi, 3\pi/2 \), and \( 2\pi \). For \( \theta = 0 \), \( r = \frac{1}{2} + 1 = \frac{3}{2} \). For \( \theta = \pi \), \( r = \frac{1}{2} - 1 = -\frac{1}{2} \), indicating the presence of the inner loop. Complete the graph around these critical values, noticing how it loops back inside for certain \( \theta \).

Step 3: Sketch Part b: Limacon with Sine

Graph the equation \( r = \frac{1}{2} + \sin \theta \) similarly. Start with key points: \( \theta = 0, \pi/2, \pi, 3\pi/2, 2\pi \). For instance, for \( \theta = \pi/2 \), \( r = \frac{1}{2} + 1 = \frac{3}{2} \), and for \( \theta = 3\pi/2 \), \( r = \frac{1}{2} - 1 = -\frac{1}{2} \), showing the inner loop. Use these points to draw the overall shape and demonstrate the looping feature.

Step 4: Full Graph Interpretation

For part a, the limacon will have its larger loop oriented horizontally along the polar axis (rightward), with the inner loop touching the origin depending on the plotted points. For part b, the larger loop is oriented vertically, due to the sine function's influence, again touching the origin for inner loop inspection.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points on a plane. Unlike Cartesian coordinates, which use (x, y), polar coordinates are written as (r, \( \theta \)), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis. This system is particularly useful for plotting curves that have a circular or spiral shape, such as circles, spirals, and limacons.

The key advantage is the simplicity in defining such curves mathematically. In our example, given the equation \( r = \frac{1}{2} + \cos \theta \) or \( r = \frac{1}{2} + \sin \theta \), each point on the limacon curve is a result of varying \( \theta \). The graph is complete once \( \theta \) runs from 0 to \( 2\pi \).

The beauty of polar coordinates lies in their ability to effortlessly describe the symmetry and periodicity of curves.

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, are crucial in polar equations. These functions oscillate between -1 and 1, making them ideal for constructing periodic curves like limacons.

In polar equations, \( \cos \theta \) and \( \sin \theta \) dictate the bending and twisting of the curve. In our exercises, adding \( \cos \theta \) or \( \sin \theta \) to \( \frac{1}{2} \) modifies the radius \( r \), directly affecting the shape.

For \( r = \frac{1}{2} + \cos \theta \), the cosine part leads to a limacon extending outward when \( \cos \theta \) is positive, and looping inward when \( \cos \theta \) is negative. Similarly, \( r = \frac{1}{2} + \sin \theta \) shapes the curve, resulting in an inner loop when \( \sin \theta \) is negative at certain points of \( \theta \).

Understanding these functions allows us to accurately predict and sketch curves, seeing how they twist and loop.

Graph Interpretation

Interpreting a limacon graph involves recognizing the distinct features and symmetry of the curve. In both examples given, the limacons have an inner loop, as indicated by \( b > a \) in the polar equation.

The sketching begins by determining key points like \( \theta = 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \). These values offer insights into the maximum and minimum radii, where the positive radius shows the outer loop and any negative radius signifies the inner loop.

For instance, at \( \theta = \pi \) in \( r = \frac{1}{2} + \cos \theta \), the negative radius creates a loop that extends inside, demonstrating the inner loop clearly.

Moreover, the orientation of the loops helps in distinguishing the curve's behavior: horizontal for cosine-based limacons and vertical for sine. Thus, graph interpretation aids in understanding both the geometry and spatial orientation of limacons.

Inner Loop

A distinctive feature of some limacons is the inner loop. This occurs when \( b > a \) in the equation \( r = a + b \cos\theta \) or \( r = a + b \sin\theta \). Essentially, the inner loop is a smaller, nested shape within the larger loop created by negative values of \( r \) for certain \( \theta \).

The limacons in our exercises demonstrate this by showing an internal loop when values like \( \theta = \pi \) or \( 3\pi/2 \) are tested. Here, \( r \) becomes negative, causing the curve to loop back through the pole (the origin), thus forming a distinct and closed smaller loop.

This inner loop represents a range where the component \( \cos \theta \) or \( \sin \theta \) pulls the curve inward rather than outward. Understanding this feature is crucial for predicting the curve's complete behavior and accurately representing it on polar graph paper.