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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.2.40
Chapter 12, Problem 12.2.40

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = 3 csc θ

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1
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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Key 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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Intro to Polar Coordinates

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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Verifying Trig Equations as Identities
Related Practice
Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola with focus at (3, 0)

Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 4 cos t, y = 4 sin t; slope = 1/2

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


25y² - 4x² = 100

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Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside one leaf of r = cos 3θ

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Textbook Question

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

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Textbook Question

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = x²