Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
(x - 1)² + y² = 1
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16. Parametric Equations & Polar Coordinates
Polar Coordinates
3:25 minutesProblem 12.2.40
Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 3 csc θ
Verified step by step guidance1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
Given the equation \(r = 3 \csc \theta\), rewrite \(\csc \theta\) in terms of sine: \(\csc \theta = \frac{1}{\sin \theta}\), so the equation becomes \(r = \frac{3}{\sin \theta}\).
Multiply both sides of the equation by \(\sin \theta\) to get \(r \sin \theta = 3\).
Use the polar-to-Cartesian conversion \(r \sin \theta = y\) to rewrite the equation as \(y = 3\).
Interpret the Cartesian equation \(y = 3\): this represents a horizontal line located 3 units above the x-axis.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar and Cartesian Coordinate Systems
Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use x and y values. Understanding how these systems relate is essential for converting equations between them, as each system describes points in the plane differently.
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Intro to Polar Coordinates
Conversion Formulas Between Polar and Cartesian Coordinates
The key formulas for conversion are x = r cos θ and y = r sin θ. These allow expressing polar equations in terms of x and y by substituting r and θ, enabling the transformation of polar curves into Cartesian form for easier analysis.
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Trigonometric Identities and Curve Description
Using identities like csc θ = 1/sin θ helps rewrite the given equation. After conversion, recognizing the resulting Cartesian equation (e.g., a line, circle, or parabola) is crucial to describe the curve's shape and properties accurately.
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7:17Verifying Trig Equations as Identities
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