Problem 61 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=1-3 \cos \theta $$

Step-by-Step Solution

Answer

The polar equation \( r = 1 - 3 \, \text{cos} \, \theta \) is a limaçon with an inner loop. Plot key points and connect them smoothly.

Step 1: Identify the Type of Polar Equation

The given equation is in the form \( r = a + b \, \text{cos} \, \theta \). This is a type of limaçon curve. Comparing it with the standard form, identify the values of \(a\) and \(b\). Here, \(a = 1\) and \(b = -3\).

Step 2: Determine the Shape of the Limaçon

For the limaçon curve in the form \(r = a + b \, \text{cos} \, \theta\):- If \(|a| < |b|\), the graph will be a limaçon with an inner loop.- Here, \(|1| < |-3|\), so the graph will have an inner loop.

Step 3: Find Key Points

Evaluate the equation for various values of \( \theta \) to find key points:- For \( \theta = 0 \), \( r = 1 - 3 \, \text{cos} \, 0 = 1 - 3 = -2 \).- For \( \theta = \frac{\text{π}}{2} \), \( r = 1 - 3 \, \text{cos} \, \frac{\text{π}}{2} = 1 - 3 \, \times 0 = 1 \).- For \( \theta = \text{π} \), \( r = 1 - 3 \, \text{cos} \, \text{π} = 1 - 3 \, \times (-1) = 4 \).- For \( \theta = \frac{3\text{π}}{2} \), \( r = 1 - 3 \, \text{cos} \, \frac{3\text{π}}{2} = 1 - 3 \, \times 0 = 1 \).

Step 4: Plot the Points

Plot the points: \(( \theta = 0, r = -2 )\), \(( \theta = \frac{\text{π}}{2}, r = 1 )\), \(( \theta = \text{π}, r = 4 )\), and \(( \theta = \frac{3\text{π}}{2}, r = 1 )\) in polar coordinates.

Step 5: Sketch the Curve

Connect the points smoothly to draw the limaçon with an inner loop. The curve will pass through these points as it loops back into itself.

Key Concepts

limaçon curves

A limaçon is a type of polar curve that can be quite fascinating. It's defined by equations of the form: \( r = a + b \text{cos} \theta \) or \( r = a + b \text{sin} \theta \). The shape of the limaçon depends on the values of \( a \) and \( b \).

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• If \( |a| < |b| \), the limaçon has an inner loop.
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• If \( |a| = |b| \), the limaçon becomes a cardioid, which looks like a heart.
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• If \( |a| > |b| \), the limaçon resembles a distorted circle called a dimpled limaçon or a limaçon without a loop if \( |b| \) is much smaller.

In the given equation \( r = 1 - 3 \text{cos} \theta \), we see that \( |1| < |-3| \), which means the graph is a limaçon with an inner loop.

polar coordinates

Polar coordinates are another way to represent points in a plane. Instead of using \( (x, y) \) like Cartesian coordinates, polar coordinates use \( (r, \theta) \), where:

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

Using polar coordinates, you can describe curves and shapes that look complex in Cartesian coordinates more simply. For example, circles, spirals, and limaçons are often easier to understand and graph in polar coordinates.

graphing polar equations

Graphing polar equations involves plotting points based on their polar coordinates \( (r, \theta) \). Here are some steps to follow:

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• Identify key points by substituting different values of \( \theta \) into the equation to find \( r \).
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• Plot each point \( (r, \theta) \) on the polar grid, where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle.
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• Connect the plotted points smoothly to visualize the curve.

For example, for the polar equation \( r = 1 - 3 \text{cos} \theta \), we can evaluate it at different angles (\( \theta = 0, \frac{\text{π}}{2}, \text{π}, \frac{3\text{π}}{2} \)) and find the corresponding \( r \) values. This gives key points to plot and helps sketch a smooth curve, revealing the inner loop of the limaçon.