Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(r=4+2 \cos \theta\) is a dimpled limaçon.

Step-by-Step Solution

Answer

The statement is false. The graph of \(r=4+2 \cos \theta\) forms a Convex Limaçon, not a Dimpled Limaçon.

Step 1: Understanding Limaçon Shapes

The form of a limaçon is \(r=a ± b \cos(θ)\) or \(r=a ± b \sin(θ)\). Next, we ascertain what makes a limaçon either dimpled, convex, or a cardioid based on the relationship between a and b. If \(| a / b | < 1\), then the limaçon is dimpled. If \(| a / b | = 1\), it is a cardioid. If \(| a / b | > 1\), the limaçon is convex.

Step 2: Applying the Limaçon Rules to the Exercise

In our equation \(r=4+2 \cos \theta\), a is 4 and b is 2. Thus, we must compare |4/2|. Which is 2.

Step 3: Determine Limaçon Type

Given that |a/b|=2, this is greater than 1. Hence, by our rules laid out above, this indicates that \(r=4+2 \cos \theta\) is a Convex Limaçon, not a Dimpled Limaçon.

Key Concepts

Limaçon

A limaçon, pronounced "lee-mah-sohn," is a type of polar curve that is interesting due to its diverse shapes. This curve can exhibit various forms depending on its mathematical parameters. The general equations for a limaçon are given by:

• \( r = a \pm b \cos(\theta) \)
• \( r = a \pm b \sin(\theta) \)

Limaçons can take different shapes based on the ratio \( |a/b| \):

• If \( |a/b| < 1 \), it is a "dimpled limaçon".
• If \( |a/b| > 1 \), it becomes a "convex limaçon".
• If \( |a/b| = 1 \), it is known as a "cardioid".

These properties make limaçons versatile in polar graphing, transitioning smoothly between these shapes by merely adjusting the values of \( a \) and \( b \). This adaptability is what makes them a fascinating topic in polar coordinate discussions.

Graphing Polar Equations

Graphing polar equations offers a unique way to visualize mathematical functions using polar coordinates. Unlike the traditional Cartesian coordinate system, polar coordinates involve two variables: the radius \( r \) and the angle \( \theta \). These variables make it possible to describe a point's location based on its distance from the origin and its angular position relative to a fixed direction.
To graph a polar equation like \( r = 4 + 2\cos(\theta) \), follow these steps:

• For several values of \( \theta \) (usually between 0 and 2\( \pi \)), calculate \( r \).
• Plot these points on polar graph paper, where the angle \( \theta \) corresponds to the direction from the pole (origin), and \( r \) is the distance from the pole.
• Connect the plotted points to see the curve's shape.

This approach allows you to see patterns and special features, such as loops or dimples, that are not as evident in the Cartesian system. Mastering graphing polar equations can enhance one's understanding of the relationship between algebraic forms and their graphical representations.

Trigonometric Functions

Trigonometric functions are a crucial element in graphing polar equations, as they dictate the behavior of polar curves like limaçons. In the context of polar equations, functions such as \( \cos(\theta) \) and \( \sin(\theta) \) determine how the radius \( r \) changes as the angle \( \theta \) varies.
Here are some key trigonometric functions used in polar coordinates:

• \( \cos(\theta) \) is responsible for horizontal shifts, affecting the curve's symmetry around the x-axis.
• \( \sin(\theta) \) influences vertical shifts, impacting symmetry around the y-axis.
• The amplitude of these trigonometric functions, given by the coefficient \( b \), will affect how pronounced these shifts appear.

By understanding these trigonometric effects, one gains insight into predicting and analyzing the curves generated by polar equations. This knowledge is fundamental not only for graphing but also for unraveling the intricate patterns in mathematical models.