Question

Identify and graph each polar equation. $$ r=4-2 \cos \theta $$

Short answer

The given polar equation \(r = 4 - 2 \cos \theta\) represents a limaçon without an inner loop.

Step-by-step solution

Understand the Polar Equation

The given polar equation is of the form \(r = a - b \cos \theta\), where \(a = 4\) and \(b = 2\). This type of polar equation represents a limaçon.

Identify the Type of Limaçon

For \(r = a - b \cos \theta\), if \(a > b\), it is a limaçon with an inner loop. Since \(4 > 2\), the graph will be a limaçon without an inner loop.

Find Key Points

Determine a few key points by substituting common angles \( \theta \) into the equation, such as \(\theta = 0\), \(\theta = \pi/2\), \(\theta = \pi\), and \(\theta = 3\pi/2\). For instance: \ \ \ \ theta = 0: \ r = 4 - 2 \cos(0) = 2 \ \ \ \ \theta = \frac{\pi}{2}: \ r = 4 \ \ \ \ \theta = \pi: \ r = 6 \ \ \ \ \theta = \frac{3\pi}{2}: \ r = 4.

Plot the Points

Use the polar coordinate system to plot the points found in Step 3: \( (2, 0) \, \ (4, \frac{\pi}{2}) \, \ (6, \pi) \, \ (4, \frac{3\pi}{2}) \).

Draw the Limaçon

Connect the points smoothly to sketch the limaçon. The curve will start at \(r = 2\) when \ \theta = 0\ and will have the largest radius of \(r = 6\) when \ \theta = \pi\. Ensure the curve accurately reflects the shape based on these key points.

Key concepts

Graphing Polar Equations

Graphing polar equations can seem challenging at first, but once you understand the fundamentals, it's quite rewarding. Polar equations use polar coordinates \(r\), the distance from the origin, and \(\theta\), the angle from the positive x-axis. Instead of the x and y coordinates, polar graphs use circles and angles to represent points. To graph a polar equation such as \(r=4-2 \cos \theta\), you start by understanding its form and identifying key points. By substituting various \(\theta\) values (like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\), you obtain corresponding \(r\) values, which you then plot on the polar coordinate system. Connecting these points smoothly reveals the polar graph's shape, such as a limaçon in this example.

Limaçon

Limaçons are a particular type of polar graph characterized by their distinctive shape. The general form of a limaçon equation is \(r = a \pm b \cos \theta\) or \(r = a \pm b \sin \theta\). Depending on the values of \(a\) and \(b\), a limaçon can manifest different features:

• If \(a > b\), the limaçon resembles a heart shape (without an inner loop).
• If \(a = b\), the limaçon forms a cardioid.
• If \(a < b\), it has an inner loop.

In our equation \(r=4-2 \cos \theta\), since \(4 > 2\), it forms a limaçon without an inner loop. This graph has distinctive characteristics such as a dimpled or oval-like outward curve.

Polar Coordinates

Polar coordinates provide a unique way to plot points using angles and distances from a central point (the origin). Instead of the Cartesian coordinate system's rectangular layout, polar coordinates describe a point by \(r\) (its radius from the origin) and \(\theta\) (the angle from the positive x-axis). This system is particularly useful for graphing circles, spirals like the Archimedean spiral, and other shapes like limaçons.
When dealing with polar equations:

• Determine the radius \(r\) for different angles \(\theta\) to establish key points.
• Use these points to create the overall graph.
• Connect points smoothly, respecting the pattern implied by the equation.

This approach allows us to visualize complex shapes in a way that is often more intuitive than using Cartesian coordinates.