Problem 22 - 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. Cardioids $$ \text { a. }r=1-\cos \theta \quad \text { b. } r=-1+\sin \theta $$

Step-by-Step Solution

Answer

Both equations form heart-shaped curves called cardioids.

Step 1: Identify Limaçon Type

Limaçon equations have the form \(r = a \pm b \cos \theta\) or \(r = a \pm b \sin \theta\). For these exercises, we have two equations provided: \(r = 1 - \cos \theta\) and \(r = -1 + \sin \theta\). These are in the forms \(r = a - b \cos \theta\) and \(r = a + b \sin \theta\), respectively, where the constants are both equal to 1. This results in a special type of limaçon called a cardioid.

Step 2: Analyze the Equation r=1−cosθ

In the equation \(r = 1 - \cos \theta\), set \(r = a - b\cos\theta\) with \(a = 1\) and \(b = 1\). Substituting \(\theta = 0\), we get \(r = 1 - 1\cos 0 = 0\). At \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\), \(\cos \theta = 0\), yielding \(r = 1\). At \(\theta = \pi\), \(\cos \pi = -1\), achieving \(r = 2\). This shape hits the pole (origin) and looks like a heart sitting on its tip.

Step 3: Graph r=1−cosθ

To graph \(r = 1 - \cos \theta\), plot the calculated points: from above, \(\theta = 0\) gives \(r = 0\), \(\theta = \frac{\pi}{2}\) gives \(r = 1\), \(\theta = \pi\) gives \(r = 2\), and \(\theta = \frac{3\pi}{2}\) returns to \(r = 1\). The graph is symmetric about the horizontal axis, creating a cardioid shape.

Step 4: Analyze the Equation r=−1+sinθ

For \(r = -1 + \sin \theta\), consider \(r = a + b\sin\theta\) with \(a = -1\) and \(b = 1\). When \(\theta = \frac{\pi}{2}\), \(\sin \theta = 1\), giving \(r = 0\). At \(\theta = \pi\) and \(\theta = 0\), where \(\sin \theta = 0\), \(r = -1\). At \(\theta = \frac{3\pi}{2}\), \(\sin \theta = -1\), \(r = -2\). This also forms a cardioid that is symmetric, primarily shifted downward.

Step 5: Graph r=−1+sinθ

Graphing \(r = -1 + \sin \theta\) involves using the points: \(\theta = 0\) gives \(r = -1\), \(\theta = \frac{\pi}{2}\) gives \(r = 0\), \(\theta = \pi\) also results in \(r = -1\), and \(\theta = \frac{3\pi}{2}\) results in \(r = -2\). The graph forms a cardioid shape, symmetric on the vertical axis, flipped and positioned downwards compared to the conventional cardioid.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This system is incredibly useful for plotting complex graphs and understanding curves like the limaçon. Instead of using \(x, y\) coordinates as in the Cartesian system, we use \(r, \theta\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle from the positive x-axis.

Here are a few important details about polar coordinates:

• The reference point, often called the pole, is similar to the origin in the Cartesian system.
• The angle is measured in radians or degrees, with positive angles measured counterclockwise.
• Graphs of polar equations can demonstrate symmetry not seen in Cartesian graphs.

Polar coordinates are particularly effective for graphing circles, spirals, and other figures like the limaçon. This system captures the fluid nature of curves which are otherwise challenging to express with rectangular coordinates.

Cardioid Shape

The cardioid is a special type of limaçon, which you might visualize as a heart shape. The name 'cardioid' comes from the Greek word for heart. It features one cusp, giving it that distinct heart-like form. Understanding cardioids in polar form involves certain standard equations, such as \(r = a \pm b \cos \theta\) or \(r = a \pm b \sin \theta\).

Key features of cardioids include:

• When \(a = b\), the cardioid has a singular point known as a cusp.
• The graph is symmetric around the axis of the trigonometric function used (cosine or sine).
• Its pole is touched by the curve, creating unique points as angles vary.

These shapes arise naturally in various physical phenomena, such as in acoustics and optics, making them particularly interesting to study. You can also see their symmetry and dynamic nature by graphing them using polar coordinates.

Trigonometric Functions

Trigonometric functions play a crucial role in defining and graphing limaçons. They're the core mathematical operations involved in polar equations with attributes of angles and distances. The primary functions are sine (\sin \theta) and cosine (\cos \theta), often used in expressing periodic motions and waves.

Here's why trigonometric functions are important:

• They define the relationship between angles and side lengths in right triangles, which is foundational in polar graphs.
• In polar equations like \(r = a \pm b \cos \theta\) or \(r = a \pm b \sin \theta\), cosine and sine dictate the cardioid symmetry.
• The periodic nature of these functions helps in visualizing cyclic and repeating patterns found in limaçons.

Understanding the behavior of these functions, especially their peaks and zero-crossings, aids in accurately sketching the graph of a limaçon. These insights are essential as they drive the underlying structure of many naturally occurring patterns.