Math Problem Statement
sketch the curves in polar coordinates
Solution
Let's examine the two polar equations and analyze how to sketch them.
(ii)
This is the polar equation of a limacon with a sine term. Here's how it works:
- When , , so the curve passes through the origin.
- When , , so the point is plotted on the positive y-axis.
- When , , so it returns to the origin.
- When , , which indicates a point in the negative y-direction.
This gives a limacon shape without an inner loop, symmetric with respect to the vertical axis.
(iii)
This is also a polar equation, and it generates a limacon with a negative and cosine term. Here's the interpretation:
- When , , so the point is 2 units to the left of the origin.
- When , , the point is 1 unit below the origin.
- When , , meaning the curve passes through the origin.
- When , , a point below the origin.
This results in a limacon with an inner loop, symmetric with respect to the horizontal axis.
Would you like me to generate an exact sketch of these curves for you?
Here are 5 related questions:
- How does the graph of differ from ?
- What happens to the curve when you adjust the coefficient in ?
- How do we determine the symmetry of a polar curve based on the equation?
- What effect does a negative coefficient have in polar equations like ?
- Can you explain how to convert a polar equation to Cartesian coordinates?
Tip: When plotting polar curves, always start by determining key points for special angles like to understand the general shape of the graph.
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Math Problem Analysis
Mathematical Concepts
Polar Coordinates
Limacon Curves
Trigonometric Functions
Formulas
r = 6sin(θ)
r = -1 - cos(θ)
Theorems
Symmetry of polar curves
Behavior of limacons based on trigonometric equations
Suitable Grade Level
Grades 10-12
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