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Magazine Chapter 8: Problem 257
For the following exercises, graph the polar equation. Identify the name of the shape. $$ r=2+2 \cos \theta $$
Short Answer
Expert verified
The graph is a cardioid.
Step by step solution
01
Identify the Type of Polar Equation
The given polar equation is \( r = 2 + 2 \cos \theta \). This equation is of the form \( r = a + b \cos \theta \), which is a type of "limacon." The constants \( a \) and \( b \) are both equal to 2.
02
Determine the Specific Shape
In our equation, \( a = 2 \) and \( b = 2 \). For a limacon of the form \( r = a + b \cos \theta \), if \( a = b \), the graph is a "cardioid." Since \( a = b \) in this problem, the shape is indeed a cardioid.
03
Plot Key Points
Consider values of \( \theta \) to understand the shape better. For example, when \( \theta = 0 \), \( r = 2 + 2\cdot1 = 4 \). When \( \theta = \pi \), \( r = 2 + 2\cdot(-1) = 0 \). These points help identify that the curve comes back to the pole (origin) at \( \pi \).
04
Sketch the Full Graph
Use symmetry properties and plot a few more points for a complete view. The nature of cosine implies that the graph is symmetric about the polar axis. By plotting points for a range of \( \theta \) values, it is clear that the shape loops around itself without forming any inner loops, characteristic of a cardioid.
05
Finalize Graph and Name Shape
Confirming from the plotted points and the path taken by the graph, it forms a heart-shaped loop. Therefore, the shape is a "cardioid".
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limacon
A limacon is a specific type of curve in polar coordinates that can appear in a variety of shapes, ranging from a looped figure to a dimpled or even a perfectly round shape. The most common form is given by the equation \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Here, \( a \) and \( b \) are constants that significantly influence the shape of the limacon.
The shape of the limacon depends heavily on the ratio between \( a \) and \( b \):
These curves are fascinating because they illustrate how changes in an equation can produce wholly different visual results.
The shape of the limacon depends heavily on the ratio between \( a \) and \( b \):
- If \( a = b \), the limacon becomes a cardioid, a special, heart-shaped curve.
- If \( a > b \), the limacon looks more like a rounded, dimpled shape.
- If \( b > a \), the limacon will have an inner loop.
These curves are fascinating because they illustrate how changes in an equation can produce wholly different visual results.
Cardioid
The cardioid is a special type of limacon. If you've ever seen a heart squeezed at the top, you've got the right image in mind. This shape is formed when \( a = b \) in the limacon equation. For our focus equation, \( r = 2 + 2 \cos \theta \), both \( a \) and \( b \) are 2, creating a perfectly symmetrical cardioid. Cardioids have a characteristic cusp at the pole (origin) where the graph touches itself.When drawing or understanding a cardioid:
- Consider the points and angles at which the shape is defined. These are often critical in plotting.
- The symmetry of the cardioid is often aligned along the horizontal or vertical axis, depending on if cosine or sine is used.
- Expect the graph to loop back to the origin at specific points like \( \theta = \pi \).
Cosine Function
The cosine function is one of the pivotal trigonometric functions and plays a crucial role in polar equations. In polar graphing, cosine determines the symmetry and orientation of the graph. When you see a polar equation like \( r = 2 + 2 \cos \theta \), the cosine term influences how the curve shifts horizontally. As only the cosine function is involved, the graph will be symmetric about the horizontal polar axis (the x-axis equivalent in polar coordinates).A few key aspects of cosine in polar graphs:
- Cosine values range from -1 to 1, affecting the maximum and minimum values of \( r \).
- At \( \theta = 0 \), \( \cos \theta = 1 \), giving the maximum value of \( r \).
- At \( \theta = \pi \), \( \cos \theta = -1 \), which often leads \( r \) back to or through the origin, indicating various graph points.
Graphing Techniques
Graphing polar coordinates requires a slightly different technique than graphing Cartesian coordinates, mainly due to their angle-radius dependency. When graphing a polar equation, such as \( r = 2 + 2 \cos \theta \), start by identifying key points and angles.Here are some core techniques to enhance graphing skills in polar coordinates:
- Identify special angles like 0, \( \pi/2 \), \( \pi \), and \( 3\pi/2 \) where the trigonometric functions take on significant values.
- Analyze how changes in \( \theta \) affect \( r \), particularly looking at symmetry and periodic behavior.
- Plot key points for calculated \( \theta \) values to get a reliable outline for the overall shape.
- Use symmetry properties to reduce the plotting effort; for cosine-related graphs, examine symmetry about the polar axis.
