Problem 43 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=2+\sin \theta$$

Step-by-Step Solution

Answer

The graph of the equation \(r=2+\sin \theta\) is a limaçon. It swings outward and inward as \(\theta\) varies from 0 to \(2\pi\). It starts at the point (2,0) and loops around the origin, reaching a maximum radius of 3 and a minimum radius of 1, forming a limaçon.

Step 1: Recognize the form of the equation

The polar equation \(r=2+\sin \theta\) is an example of a limaçon. When the equation is in the form \(r=a + bsin \theta\) or \(r=a + bcos \theta\), the graph is a limaçon. Here we can see that a and b are both 1, so our graph of the equation will be a limaçon.

Step 2: Plot key points

As \(\theta\) ranges from 0 to \(2\pi\), we calculate the corresponding values of r. For example, at \(\theta=0, r=3\); at \(\theta=\pi/2, r=3\); at \(\theta=\pi, r=1\); and at \(\theta=3\pi/2, r=3\). These points will help us to start sketching the graph.

Step 3: Sketch the graph

Using the points from Step 2, the next step is to sketch the limaçon. The graph starts at r=2 (the point (2,0)) when \(\theta=0\), moves outward as \(\theta\) approaches \(\pi/2\), moves back inward as \(\theta\) approaches \(\pi\), moves outward again as \(\theta\) approaches \(3\pi/2\), and finally returns to the starting point as \(\theta\) approaches \(2\pi\). Notice that the graph loops around the pole (the point at the origin).

Key Concepts

Understanding Limaçons

The term limaçon comes from the Latin word limax, meaning snail, which represents the shape of the curve in some cases. Limaçons are a family of polar curves that resemble a circle, but with a section that has been pulled or pushed away from the center, creating a distorted loop or dimple.

A standard form for the equation of a limaçon is either \(r=a + b\sin \theta\) or \(r=a + b\cos \theta\), where a and b are constants. The shape and size of the limaçon depend on the relationship between a and b.

• If a = b, the limaçon has an inner loop.
• If a > b, the limaçon has a dimpled shape.
• If a < b, it is convex with no inner loop.

In the exercise \(r=2+\sin \theta\), a and b are both equal to 1. This means the graph will display a slight dimple, characteristic of a limaçon where a = b. Sketching such graphs requires plotting key points as \(\theta\) varies, and then connecting these points with a smooth curve that captures the determining feature of a limaçon - its distorted loop or dimple.

Plotting Polar Equations

When it comes to polar equations, the goal is to express the relationship between the radius r and the angle \(\theta\) in the polar coordinate system. Plotting the graph of a polar equation involves calculating the radius for various angles and then drawing the corresponding points on a polar grid.

For the given equation, \(r=2+\sin \theta\), you can compute the radius for significant angles, such as \(\theta=0\), \(\theta=\frac{\pi}{2}\), \(\theta=\pi\), and \(\theta=\frac{3\pi}{2}\). After plotting these and other points, you draw a curve through them to represent the graph. This method can be applied to any polar equation, offering a visual understanding of the relationship between r and \(\theta\).

Navigating Polar Coordinates

The polar coordinate system is an alternative to the traditional Cartesian system for defining positions on a plane. Instead of using x and y coordinates, polar coordinates specify a point's location based on its distance from a central point, called the pole, and the angle formed with the positive x-axis, known as the polar axis.

Polar coordinates are written as (r, \(\theta\)), where r is the radial distance from the pole, and \(\theta\) is the angle measured in radians from the polar axis. In our exercise, points on the graph of \(r=2+\sin \theta\) are determined by calculating the value of r for various angles \(\theta\). This approach offers a unique way to explore curves like limaçons, and it is particularly useful for shapes that are symmetric with respect to a central point.