Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar equation \(r = 2 - \cos \theta\) is a limacon with an inner loop.

Step 1 - Identify the Polar Equation

Recognize that the given equation is in the polar form: \[ r = 2 - \cos \theta \].

Step 2 - Recognize the Type of Polar Graph

Identify the type of curve based on the polar equation. The given equation is of the form \[ r = a - b \cos \theta \].This is the general form of a limacon.

Step 3 - Determine Graph Attributes

Compare the given equation \[ r = 2 - \cos \theta \] with the standard limacon form \[ r = a - b \cos \theta \]. Here we see that \(a = 2\) and \(b = 1\). Since \(a > b\), the limacon will have an inner loop.

Step 4 - Plot Key Points

Calculate a few key points to help plot the graph. For example, compute \(r\) for \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\):\(\theta = 0\), \(r = 2 - \cos(0) = 1\)\(\theta = \frac{\pi}{2}\), \(r = 2 - \cos\left(\frac{\pi}{2}\right) = 2\)\(\theta = \pi\), \(r = 2 - \cos(\pi) = 3\)\(\theta = \frac{3\pi}{2}\), \(r = 2 - \cos\left(\frac{3\pi}{2}\right) = 2\)

Step 5 - Sketch the Graph

Use the calculated points and the symmetry of the limacon to sketch the graph. The curve will form a limacon with a slight inner loop.

Key Concepts

limacon curve

Let's dive into what a limacon curve is. The limacon (pronounced 'lee-muh-kon') is a special kind of polar curve that has a unique shape. It can have an inner loop, a dimple, or be almost circular, depending on its equation. A limacon curve is generally described by the polar equation: \[r = a - b \cos \theta\] or \[r = a - b \sin \theta\]. Here, \a\ and \b\ are constants that determine the shape of the curve.
One example of the limacon curve is given by the equation: \[ r = 2 - \cos \theta\]. This is the equation used in our exercise. By comparing it with \[ r = a - b \cos \theta\], we see that \a = 2\ and \b = 1\.
Because \a > b\, this limacon will feature an inner loop. These curves are fascinating because of their variety and unique shapes.

plotting polar graphs

Plotting polar graphs requires understanding how to translate polar coordinates into points on a graph. In polar coordinates, each point is determined by an angle \ \theta\ \ and a radius \ r \. Let's break down how to plot polar graphs step-by-step:
First, identify the polar equation and its type. Here, we identified \[ r = 2 - \cos \theta\]\ as a limacon curve. Next, calculate key points by plugging in different \ \theta\ \ values, for instance \ \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\ \. You get different \ \r\ \ values such as \1\, \2\, \3\, and \2\, respectively.
Then, plot these points on polar graph paper. Remember each pair (\theta, r) represents a point. By connecting these points smoothly, you get the shape of the graph. For the limacon, you'd see its distinctive inner loop. The more points you calculate, the more accurate your graph will be.

polar coordinates

Polar coordinates are a way of locating points in a plane using the distance from a fixed point (the origin) and an angle from a fixed direction (usually the positive x-axis). More formally, each point in the plane is given by a pair (\theta, r):

• \ \theta\ (theta) is the angle measured from the positive x-axis.
• \ \r\ (radius) is the distance from the origin to the point.

To better understand, let's look at a simple example: If \[ \theta = \frac{\pi}{2}\]\ and \[ \r = 3\]\, this means the point is 3 units away from the origin, measured at an angle of \90\ degrees from the positive x-axis (or \ \frac{\pi}{2}\ radians). This system is very useful for dealing with circular and spiral patterns that occur frequently in nature and physics.
Understanding polar coordinates is essential for effectively plotting and interpreting polar graphs, like the limacon curve.