Problem 23 - 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. Dimpled limaçons $$ \text { a. r=\frac{3}{2}+\cos \theta \quad \text { b. } r=\frac{3}{2}-\sin \theta $$

Step-by-Step Solution

Answer

Plot both equations in polar coordinates and observe the dimpled shapes.

Step 1: Identify the Type of Limaçon

The given equations are both in the form of a limaçon. Specifically, the equations are of the form \( r = a + b\cos\theta \) and \( r = a - b\sin\theta \). For these equations, if \( |a| = |b| \), the limaçon will have a dimpled shape.

Step 2: Graph Equation a: r=\frac{3}{2}+\cos \theta

For the equation \( r = \frac{3}{2} + \cos \theta \), we start by considering the key points: - When \( \theta = 0 \), \( r = \frac{3}{2} + 1 = \frac{5}{2} \).- When \( \theta = \frac{\pi}{2} \), \( r = \frac{3}{2} + 0 = \frac{3}{2} \).- When \( \theta = \pi \), \( r = \frac{3}{2} - 1 = \frac{1}{2} \).- When \( \theta = \frac{3\pi}{2} \), \( r = \frac{3}{2} + 0 = \frac{3}{2} \).Plot these points in polar coordinates and draw the resulting limaçon, noting the dimple on the side of the positive x-axis.

Step 3: Graph Equation b: r=\frac{3}{2}-\sin \theta

For the equation \( r = \frac{3}{2} - \sin \theta \), we analyze the key points:- When \( \theta = 0 \), \( r = \frac{3}{2} - 0 = \frac{3}{2} \).- When \( \theta = \frac{\pi}{2} \), \( r = \frac{3}{2} - 1 = \frac{1}{2} \).- When \( \theta = \pi \), \( r = \frac{3}{2} + 0 = \frac{3}{2} \).- When \( \theta = \frac{3\pi}{2} \), \( r = \frac{3}{2} + 1 = \frac{5}{2} \).Again, plot these points and connect them to form the dimpled limaçon, noting the dimple on the y-axis.

Key Concepts

Understanding Limaçon in Polar Coordinates

A limaçon is a fascinating shape that arises in polar coordinates from the equations of the form \( r = a \pm b \cos \theta \) or \( r = a \pm b \sin \theta \). The name "limaçon," which means "snail" in French, is derived from the curves these equations produce. There are four primary types of limaçon based on the relationship between \( a \) and \( b \):

• Cardioid (when \( |a| = |b| \))
• Inner loop (when \( |a| < |b| \))
• Dimpled limaçon (when \( |a| \approx |b| \))
• Convex limaçon (when \( |a| > |b| \))

In the context of the given problem, understanding whether the curve is dimpled, looping, or smooth helps predict and visualize what the graph should look like. Limaçons are notable because they can create a rich variety of shapes simply by adjusting the parameters \( a \) and \( b \). This versatility makes them a perfect topic of study for exploring the behavior of polar equations.

Basics of Graphing Polar Equations

Graphing polar equations is an exciting part of mathematics that offers a different perspective from cartesian coordinates. In polar coordinates, each point on the plane is determined by a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis. When graphing polar equations like those for limaçons, there are practical steps you can follow:

• Identify key angle values where the equation simplifies, such as \( \theta = 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \).
• Calculate the corresponding \( r \) values for these angles. This will give you a set of points that outline the curve.
• Plot these points with care and draw smooth curves between them to complete the graph.

For the equations \( r = \frac{3}{2} + \cos \theta \) and \( r = \frac{3}{2} - \sin \theta \), this process involves plotting and connecting points to reveal dimpled shapes. It's a hands-on way to understand the unique nature of these shapes in polar coordinates.

Exploring Dimpled Shapes in Polar Graphs

Dimpled shapes are a specific form of limaçon where the parameter values \( a \) and \( b \) are nearly equal, giving the curve a distinct feature. Unlike other limaçon forms, dimpled limaçons showcase a small indent or 'dimple' along one side of the shape. This occurs when \( |a| \equiv |b| \).For the given equations, they illustrate dimpled shapes by placing a small indentation:

• For \( r = \frac{3}{2} + \cos \theta \), the dimple appears on the positive x-axis side.
• For \( r = \frac{3}{2} - \sin \theta \), the dimple aligns along the y-axis.

These dimples are not just distinctive; they highlight how sensitive polar graphs are to small changes in equation structure. Understanding how dimples form and where they appear can be intriguing, as it exemplifies the role of symmetry and balance in polar graphs.