Problem 48 - Algebra and Trigonometry

Question

Graph the family of polar equations \(r=1+c \sin 2 \theta\) for \(c=0.3,0.6,1,1.5,\) and \(2 .\) How does the graph change as \(c\) increases?

Step-by-Step Solution

Answer

The graph transitions from dimpled to inner-loop limaçons as \(c\) increases.

Step 1: Understanding Polar Equations

The given family of polar equations is of the form \(r = 1 + c \sin 2\theta\). This is a type of limaçon curve, which can take different shapes based on the value of \(c\). Our task is to graph these equations for different \(c\) values and observe how the graph changes.

Step 2: Setting up the Parameter Values

The parameter values provided are \(c = 0.3, 0.6, 1, 1.5,\) and \(2\). For each of these values, we will adjust the equation \(r = 1 + c \sin 2\theta\) and graph the results.

Step 3: Graphing the Equation for c=0.3

For \(c = 0.3\), the equation becomes \(r = 1 + 0.3 \sin 2\theta\). This results in a limaçon with a noticeable dimple as \(c < 1\), indicating it is an inner-loop or "dimpled" limaçon.

Step 4: Graphing the Equation for c=0.6

For \(c = 0.6\), substitute \(c\) into the equation to get \(r = 1 + 0.6 \sin 2\theta\). The graph will similarly show a dimple, but it becomes slightly more pronounced than when \(c = 0.3\).

Step 5: Graphing the Equation for c=1

When \(c = 1\), the equation becomes \(r = 1 + \sin 2\theta\), creating a cardioid shape. The graph crosses itself at the origin, forming a heart-like shape.

Step 6: Graphing the Equation for c=1.5

For \(c = 1.5\), substitute \(c\) into the equation \(r = 1 + 1.5 \sin 2\theta\). Now, \(c > 1\), creating an inner loop within the limaçon, visually distinct from the cases with \(c \leq 1\).

Step 7: Graphing the Equation for c=2

When \(c = 2\), we have \(r = 1 + 2 \sin 2\theta\). This results in a more pronounced inner loop, indicating that as \(c\) increases past 1, the inner loop expands significantly.

Step 8: Analyzing Changes as c Increases

As \(c\) increases from less than 1 to greater than 1, the graph transitions from a dimpled limaçon to a cardioid, and then to limaçons with increasingly larger inner loops. The graph's symmetry persists around the polar axis but features change markedly across the specified \(c\) values.

Key Concepts

Understanding Limaçon Curves

Limaçon curves are a fascinating type of polar graph that produce a wide variety of shapes depending on their parameters. The general form of a limaçon equation is \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \). In the exercise, our specific version is \( r = 1 + c \sin 2\theta \). Here, the shape of the curve depends largely on the value of \( c \).

Depending on \( c \), limaçon curves can appear in several forms:

• Dimpled Limaçon occurs when \( 0 < c < 1 \). The curve resembles a rounded shape with a small indentation.
• Cardioid is formed when \( c = 1 \). This shape is quite distinctive, resembling a heart. In the graph, there will be no inner or outer loop.
• Looped Limaçon happens when \( c > 1 \). Here, the curve develops a clearly visible loop.

These alterations in shape stem from how the parameter affects the equation's radial distance as the angle \( \theta \) changes.

Graphing Polar Coordinates

Polar coordinates use a point's distance from the origin, \( r \), and its angle from a reference direction, \( \theta \), to plot its position. This system is particularly useful in graphing curves like limaçons, where the position is defined in terms of angles and radii rather than Cartesian coordinates.

When graphing polar coordinates, each point on the curve is identified by:

• The angle \( \theta \), which is measured from the polar axis.
• The radial distance \( r \), which measures how far a point is from the pole (origin).

In our exercise, the equation \( r = 1 + c \sin 2\theta \) indicates that the shape is dynamic; the distance \( r \) changes based on \( \theta \). By plotting these points for various values of \( c \), we can see the transformation from dimpled to cardioid to looped limaçons.

The Cardioid Explained

The cardioid is a special case of the limaçon curve and looks like a heart. Its distinct shape emerges specifically when the parameter \( c = 1 \). In polar coordinates, its equation becomes \( r = 1 + \sin 2\theta \), which results in a curve that touches itself at the origin.

Key characteristics of a cardioid include:

• The curve passes through the origin, meaning \( r = 0 \) at times.
• It has rotational symmetry about the polar axis. This means that if you rotate the curve 180 degrees about its center, it still looks the same.

Cardioids are not just visually appealing; they also have applications in fields like acoustics and engineering, where their properties are used for sound focusing or in gear design. Observing the transition to a cardioid as \( c \) reaches 1 is an excellent example of the diversity within polar equations.