Problem 39 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Graph the limaçon with an inner loop present at \( r = 2 + 2 \, \cos \theta \).

Step 1: Recognize the Standard Polar Form

Identify the given polar equation and recognize it as part of a standard form. The equation given is \[ r = 2 + 2 \, \cos \theta \]This is a polar equation that can be identified as a limaçon.

Step 2: Determine Key Characteristics

A limaçon of the form \[ r = a + b \, \cos \theta \]The constants in the equation are: \[ a = 2 \, \text{and} \, b = 2 \]Since \[ a = b \]this specific limaçon will have an inner loop.

Step 3: Find Important Points

Find and plot the points for specific values of \( \theta \):For \( \theta = 0 \):\[ r = 2 + 2 \, \cos(0) = 4 \]For \( \theta = \frac{\pi}{2} \):\[ r = 2 + 2 \, \cos \left( \frac{\pi}{2} \right) = 2 \]For \( \theta = \pi \):\[ r = 2 + 2 \, \cos(\pi) = 0 \]For \( \theta = \frac{3\pi}{2} \):\[ r = 2 + 2 \, \cos \left( \frac{3\pi}{2} \right) = 2 \]

Step 4: Sketch the Graph

Using the points calculated from Step 3, sketch the graph of the limaçon. The plot should reflect the shape with an inner loop, starting at (4, 0) and going through (0, π).

Key Concepts

limaçon

A limaçon is a type of curve represented by polar equations of the form either \( r = a + b \, \text{cos} \, \theta \) or \( r = a + b \, \text{sin} \, \theta \). Limaçons have unique and interesting shapes that vary depending on the values of \( a \) and \( b \). In our case, \( r = 2 + 2 \, \text{cos} \, \theta \), the constants \( a \) and \( b \) both equal 2. Some key characteristics of limaçons include:

• If \( a < b \), the limaçon has an inner loop.
• If \( a = b \), it's a special limaçon with an inner loop.
• If \( a > b \), it appears without an inner loop.

In our problem, since \( a = b \), the resulting limaçon possesses an inner loop. When you plot this curve, you'll notice how it loops back towards the origin.

graphing polar coordinates

Understanding polar coordinates is essential for graphing polar equations like the limaçon. A polar coordinate system represents a point in the plane using the distance from the origin \( r \) and the angle \( \theta \) from the positive x-axis. Here’s a quick overview:

• The distance \( r \) measures how far away a point is from the origin (pole).
• The angle \( \theta \) measures the rotational offset from the positive x-axis (polar axis).

To graph our polar equation \( r = 2 + 2 \, \text{cos} \, \theta \):

• Calculate \( r \) for several values of \( \theta \), like \( 0 \, \text{and} \, \frac{\text{π}}{2}, \, \text{π}, \, \frac{\text{3π}}{2} \).
• Plot these points on the polar coordinate plane.
• Connect the points to visualize the limaçon curve.

This process lets you see the loop and inner structures by plotting key points where \( \theta \) equals 0, \( \frac{\text{π}}{2} \), π, and \( \frac{\text{3π}}{2} \). Effective graphing results in a comprehensible graph of your polar equation.

trigonometric functions

Trigonometric functions like \( \text{cos} \, \theta \) and \( \text{sin} \, \theta \) are fundamental to understanding polar equations. In our exercise, \( r = 2 + 2 \, \text{cos} \, \theta \), we rely on the cosine function to determine \( r \) for different \( \theta \) values. Let's break down their role:

• Cosine function \( \text{cos} \, \theta \) returns a value between -1 and 1.
• For \( \theta = 0 \), \( \text{cos} \, 0 = 1 \), effectively making \( r = 2 + 2 \times 1 = 4 \).
• For \( \theta = \frac{\text{π}}{2} \), \( \text{cos} \, \frac{\text{π}}{2} = 0 \), leading to \( r = 2 + 2 \times 0 = 2 \).
• For \( \theta = \text{π} \), \( \text{cos} \, \text{π} = -1 \), resulting in \( r = 2 + 2 \times -1 = 0 \).

Understanding cosine values at these key angles lets you plot points easily. Consequently, making the graphing of polar equations clear and straightforward.