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Magazine Chapter 8: Problem 262
For the following exercises, graph the polar equation. Identify the name of the shape. $$ r=7+4 \sin \theta $$
Short Answer
Expert verified
The graph is a limacon with an inner dimple called a dimpled limacon.
Step by step solution
01
Understand the Polar Equation Form
The given polar equation is \( r = 7 + 4 \sin \theta \). This equation is in the form \( r = a + b \sin \theta \), which is a type of limacon curve. This specific form indicates variations in the graph's shape based on the values of \( a \) and \( b \).
02
Identify the Values
For the equation \( r = 7 + 4 \sin \theta \), identify the values of \( a \) and \( b \). Here, \( a = 7 \) and \( b = 4 \). Compare these values to recognize the type of limacon.
03
Determine the Type of Limacon
Since \( a > b \) (\( 7 > 4 \)), the limacon is of the type with an inner loop, also known as a limacon without a loop or dimpled limacon. It doesn't have an inner loop but may appear with a dip at the pole in some orientations.
04
Plot Key Points of the Graph
To accurately graph, calculate a few key points by assigning typical angles to \( \theta \) such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). For example: - \( \theta = 0 \), \( r = 7 + 4 \sin(0) = 7 \).- \( \theta = \frac{\pi}{2} \), \( r = 7 + 4 \sin(\frac{\pi}{2}) = 11 \).- \( \theta = \pi \), \( r = 7 + 4 \sin(\pi) = 7 \).- \( \theta = \frac{3\pi}{2} \), \( r = 7 + 4 \sin(\frac{3\pi}{2}) = 3 \).
05
Draw the Graph Based on Polar Coordinates
Utilize the calculated points to draw the graph on polar coordinate paper, keeping the nature of a limacon in mind. Plot each point and connect them smoothly to reflect the curve of \( r = 7 + 4 \sin \theta \). Notice the shape bulges out more since it鈥檚 a limacon with no inner loop and has a dimple.
06
Confirm and Identify the Graph Shape
Make sure the graph reflects an approximate heart shape or circle with a dimple, which is consistent with a limacon without an inner loop based on the equation \( r = a + b \sin \theta \) where \( a > b \). Ensure all plotted points line up accurately with what calculations and known limacon forms predict.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Limacon
A limacon is a type of polar curve often exhibiting a looped or dimpled shape. Named after the Latin word for snail, limacons display fascinating forms when graphed in polar coordinates. The standard equation for a limacon is \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \). These forms are characterized by their distinctively bulging or looping nature.
- If \( |a| = |b| \), the limacon forms a cardioid, resembling a heart shape.
- If \( |a| > |b| \), it results in a limacon without a loop but with a potential dimple.
- If \( |a| < |b| \), the limacon has a pronounced inner loop.
Graphing Polar Equations
Graphing polar equations involves understanding a different coordinate system than the usual Cartesian method. In polar coordinates, a point is represented by \( (r, \theta) \), where \( r \) is the radial distance from the origin, and \( \theta \) is the angle from the positive x-axis. This method emphasizes the curve's symmetry and shape.
To graph \( r = 7 + 4 \sin \theta \):
To graph \( r = 7 + 4 \sin \theta \):
- Use polar graph paper, which has concentric circles and radiating lines representing common angles.
- Calculate and plot points at key angles (e.g., \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \)). For each angle, substitute \( \theta \) into the equation to find \( r \).
- Connect the plotted points smoothly to reveal the limacon's curve.
Plotting Polar Curves
Plotting polar curves is a fascinating visual representation where imagination meets mathematical elegance. With polar coordinates, you can trace out beautiful curves by plotting points derived from mathematical equations.
To plot a polar curve like a limacon:
To plot a polar curve like a limacon:
- Start by choosing several key angles, known as ordinates (e.g., \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \)).
- Calculate the radial distance for each angle, using the polar equation. For example, at \( \theta = \frac{\pi}{2} \), the radius is 11 for \( r = 7 + 4 \sin \theta \).
- Plot these points on polar graph paper. The intersection of the angle line (from the origin) and the radius arc defines each point.
- Connect the dots smoothly to form the full curve.
