Math Problem Statement
- Sketch the given polar curve and identify the name of this polar curve.
𝑟=3+2cos(𝜃)
Solution
The given polar equation is . This is a limacon with a cosine term.
Steps to sketch the curve:
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Understanding the shape: The equation represents a limacon. Since it has the form , the limacon will be symmetric about the polar axis (the horizontal axis in polar coordinates).
- If , the limacon does not have an inner loop.
- If , the limacon will have a sharp point (called a cardioid).
- If , the limacon will have an inner loop.
Here, and , so . This means the limacon does not have an inner loop.
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Plotting specific points:
- At : . So the point is .
- At : . So the point is .
- At : . So the point is .
- At : . So the point is .
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Plotting the curve:
- The maximum value of is 5, and the minimum value is 1. The curve will be larger on the right side (positive ) and smaller on the left side (negative ).
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Shape of the curve: Since , the curve resembles a limacon without an inner loop. The curve bulges outward but does not loop back inside.
Final Answer:
The name of the polar curve is limacon without an inner loop.
Would you like me to generate a sketch of this polar curve for you?
Here are some related questions to expand on this concept:
- What happens to the shape if in the limacon equation?
- How does changing to affect the curve?
- Can you find the area enclosed by this limacon curve?
- What type of symmetry does this curve exhibit?
- How would you convert this polar equation into Cartesian form?
Tip: Limacons can have different appearances depending on the relationship between and (inner loop, dimpled, or no loop).
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Math Problem Analysis
Mathematical Concepts
Polar Coordinates
Graphing Polar Equations
Limaçon Curves
Formulas
Polar equation: r = a + b cos(θ)
Key points: r(0) = a + b, r(π) = a - b
Theorems
Symmetry about the polar axis
Classification of Limaçons based on a and b
Suitable Grade Level
Grades 10-12
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