Problem 24 - Thomas Calculus

Question

Graph the limaçons in Exercises \(21-24 .\) Limaçon ("lee-ma-sahn") is Old French for "snail." You will understand the name when you graph the limaçons in Exercise \(21 .\) Equations for limaçons have the form \(r=a \pm b \cos \theta\) or \(r=a \pm b \sin \theta .\) There are four basic shapes. Oval limaçons $$ \text { a. }r=2+\cos \theta \quad \text { b. } r=-2+\sin \theta $$

Step-by-Step Solution

Answer

Graph an oval shape for each equation using key points; the first is above the polar axis, the second below.

Step 1: Identify the Type of Limaçon

Given the equation, we identify it as a limaçon. The equation in part a) is \( r = 2 + \cos \theta \). This fits the form \( r = a + b \cos \theta \), where \( a = 2 \) and \( b = 1 \). In part b), the equation is \( r = -2 + \sin \theta \), fitting the form \( r = a + b \sin \theta \), where \( a = -2 \) and \( b = 1 \).

Step 2: Calculation for Range of r

For the equation \( r = 2 + \cos \theta \), the range of \( \cos \theta \) is \([-1, 1]\). Thus, \( r \) ranges from \( 2 + 1 = 3 \) to \( 2 - 1 = 1 \). Similarly, for \( r = -2 + \sin \theta \), \( \sin \theta \) also ranges from \([-1, 1]\). Thus, \( r \) ranges from \(-2 + 1 = -1 \) to \(-2 - 1 = -3 \).

Step 3: Determine the Shape

Since \( a = 2 \) and \( b = 1 \) (where \( a > b \)) in part a, the limaçon is an oval shape, as there is no inner loop. In part b, where \( a = -2 \) and \( b = 1 \) (\( |a| > b \)), it suggests the graph will loop below the polar axis due to a negative "a" resulting in a symmetric figure to that above the polar axis.

Step 4: Plot Key Points for Graph

To graph \( r = 2 + \cos \theta \): Calculate key points such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), yielding \( r = 3, 2, 1, 2 \) respectively. Similarly, for \( r = -2 + \sin \theta \), key points are calculated with \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), yielding \( r = -2, -1, -2, -3 \).

Step 5: Draw the Graph

Using the key points and knowledge of the range of \( r \), draw the graph of the limaçon. For part a, plot the points and form an oval shape above the polar axis. For part b, the graph is a similar oval shape but located below the polar axis, forming a downward loop.

Key Concepts

Polar Coordinates

Polar coordinates are a mathematical system for representing points in a plane using a radius and an angle. Unlike Cartesian coordinates (x, y), polar coordinates are useful in situations where symmetry around a point is more natural or convenient. In polar coordinates, a point is described as \[ (r, \theta) \]where:

• \( r \) is the distance from the origin or pole.
• \( \theta \) is the angle measured from the positive x-axis, typically in radians.

This system is particularly useful in graphing complex shapes like circles, spirals, and limaçons. When dealing with equations like those of limaçons, understanding the behavior of \( r \) as a function of \( \theta \) is essential.
By examining how \( r \) changes with different \( \theta \) values, you can plot points and understand the resulting shape's full path. It is crucial in determining particular points along the graph, as shown in the exercise with calculations at various angles \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). This assists in sketching the curve accurately on a polar graph.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a pivotal role in the formation and graphing of limaçons. The equations given, such as \[ r = 2 + \cos \theta \] and \[ r = -2 + \sin \theta \],involve either the cosine or sine function. Understanding the properties of these functions helps to predict the nature of the limaçon's shape.
For the cosine function:

• \( \cos\theta \) varies between -1 and 1.
• It results in changes in \( r \) that produce symmetrical patterns about the \(\theta\) axis.

Similarly, for sine:

• \( \sin\theta \) also varies between -1 and 1.
• It often affects the vertical symmetry and can result in complex shapes in polar coordinates.

Recognizing these trigonometric patterns enables you to determine points’ positions relative to the pole. This helps in visualizing how the limaçon morphs in shape from oval to oblong, or even developing loops, based on the values of \( a \) and \( b \).

Curve Sketching

Curve sketching in the context of polar coordinates, particularly for limaçons, combines key mathematical concepts to effectively create the shape of the curve. By understanding and using the relationships of polar coordinates and trigonometric functions, you can create a roadmap for plotting these curves accurately.
To sketch a limaçon:

• Identify the equation form first. Whether it's \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \), it dictates the process of plotting.
• Determine the values of \( a \) and \( b \), which influence the shape.
• Calculate key points, as demonstrated in the step-by-step solutions with angles \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
• Plot these points on the polar graph.
• Connect the dots following the logical symmetry indicated by the equation, by smoothly joining points to form an accurate representation of the shape.

Curve sketching requires practice, especially with symmetry and how different trigonometric values modify the shape. By thoroughly understanding these principles, you can create precise and beautiful representations of limaçons and similar polar curves.