Math Problem Statement
Solution
The image contains two math problems:
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The first problem seems to be asking for the Cartesian coordinates corresponding to polar coordinates of different points. Specifically, it lists:
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The second problem asks to sketch the curve defined by the polar equation: and then convert this equation into Cartesian coordinates.
Solving these problems:
1. Convert Polar to Cartesian Coordinates:
To convert polar coordinates to Cartesian coordinates , we use the formulas:
Let's calculate for each point:
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a. (3, 45°): Cartesian coordinates:
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b. (2, -120°): Cartesian coordinates:
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c. (-1, 270°): Cartesian coordinates:
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d. (-2, -60°): Cartesian coordinates:
2. Sketch and Convert the Polar Equation to Cartesian Form:
To convert the polar equation to Cartesian coordinates, use the following relationships:
Given: Substitute : Multiply both sides by : Substitute : Since , replace with : This is the equation of the curve in Cartesian coordinates. It is a bit complex to sketch, but it resembles a limaçon.
Let me know if you would like more details or have any questions!
Here are some related questions:
- How do you determine which trigonometric function to use when converting polar coordinates?
- What other types of polar curves are common, besides limaçons?
- What is the significance of the negative radius in polar coordinates?
- How can we graph polar equations using a Cartesian grid?
- What happens to the graph of if you change the coefficients?
Tip: Always double-check the sign of the angle when converting from polar to Cartesian coordinates, as it impacts the direction in which the point is plotted.
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Math Problem Analysis
Mathematical Concepts
Polar coordinates
Cartesian coordinates
Trigonometry
Graphing curves
Formulas
x = r * cos(θ)
y = r * sin(θ)
r = sqrt(x^2 + y^2)
sin(θ) = y / r
Theorems
Polar to Cartesian conversion
Polar equations in Cartesian form
Suitable Grade Level
Grades 10-12
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