Problem 28 - Precalculus

Question

Graph the polar equations. $$r=1.5-\cos \theta(\text {limaçon with no inner loop })$$

Step-by-Step Solution

Answer

The graph is a dimpled limaçon with no inner loop, extending from 0.5 to 2.5 in radius.

Step 1: Understand the Polar Equation

The given polar equation is \( r = 1.5 - \cos \theta \). This is a type of limaçon. A limaçon is a graph that varies from a circle or has a loop depending on the coefficients. In our case, since \( 1.5 > 1 \), it is a limaçon without an inner loop.

Step 2: Analyze the Effects of Parameters

The equation \( r = a - b \cos \theta \) describes a limaçon. By comparing, we can identify \( a = 1.5 \) and \( b = 1 \). Since \( a > b \), the limaçon will be dimpled, indicating it has no inner loop.

Step 3: Determine the Shape of the Limaçon

Using the information that \( a > b \), we know the limaçon will be dimpled with no loop. It will start from \( r = 0.5 \) when \( \theta = 0 \) and extend to \( r = 2.5 \) when \( \theta = \pi \), creating a slightly flattened circle form.

Step 4: Plot Key Points

Compute specific values to identify key points for plotting: at \( \theta = 0 \), \( r = 1.5 - 1 = 0.5 \); at \( \theta = \pi/2 \), \( r = 1.5 \); at \( \theta = \pi \), \( r = 1.5 + 1 = 2.5 \); and at \( \theta = 3\pi/2 \), \( r = 1.5 \).

Step 5: Sketch the Graph

Begin plotting with the key points calculated: (0.5, 0), (1.5, \( \pi/2 \)), (2.5, \( \pi \)), and (1.5, \( 3\pi/2 \)). Connect these points with a smooth curve to form the limaçon, ensuring the shape reflects a dimpled form without an inner loop.

Key Concepts

Limaçon

The term "limaçon" refers to a family of curves in polar coordinates that can take on various interesting shapes, including loops and dimples. These were first studied by the mathematician Étienne Pascal, and their name comes from the French word for "snail," due to their characteristic spiral-like form.

Limaçons are typically described by equations of the form \[ r = a - b \cos \theta \] or \[ r = a - b \sin \theta \].

The specific form of the limaçon depends on the relationship between the constants \(a\) and \(b\):

• If \( a = b \), the limaçon forms a cardioid, which is heart-shaped.
• If \( a > b \), no inner loop is formed, resulting in a dimpled limaçon.
• If \( a < b \), the curve will have an inner loop.

In the exercise given, since \( a = 1.5 \) and \( b = 1 \), with \( a > b \), it confirms that the Limaçon is a dimpled one without any inner loop. Understanding these properties can help you visualize and sketch the curve efficiently.

Polar Equations

Polar equations represent curves on the polar coordinate system, where each point on the plane is determined by a distance from the origin, \(r\), and an angle \(\theta\) from the positive x-axis.

Unlike Cartesian coordinates where points are plotted by \(x\) and \(y\) values, in polar coordinates, the position is given by \( (r, \theta) \). This makes it particularly convenient for representing curves like circles, spirals, and limaçons.

To convert a polar equation to a more familiar Cartesian form, you can use the relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

However, many geometrical shapes, such as the limaçon, are more naturally and elegantly represented using their polar form.

Using polar equations can simplify graphing and understanding these curves, as well as help you see symmetries and relationships that aren't immediately obvious in Cartesian coordinates.

Graphing Techniques

Graphing polar equations like the limaçon involves recognizing key points and how they connect in a smooth manner on the polar plane.

First, identify important values of \(\theta\) at which you will calculate \(r\). These might include \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\). For each angle, substitute into the polar equation to find the corresponding \(r\). These pairs \((r, \theta)\) form the base points for your plot.

As shown in the exercise, at specific angles:

• For \(\theta = 0\), \(r = 0.5\)
• For \(\theta = \pi/2\), \(r = 1.5\)
• For \(\theta = \pi\), \(r = 2.5\)
• For \(\theta = 3\pi/2\), \(r = 1.5\)

Once these key points are plotted, connect them with a smooth curve reflection. The shape should reflect the type of limaçon you're graphing, such as dimpled in this exercise.

Techniques involve understanding symmetry, the range of \(r\) values, and connecting curves in the correct direction. By practicing with different polar equations, you will become proficient at visualizing and sketching these unique, elegant curves.