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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 12.2.49
Chapter 12, Problem 12.2.49

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


y = x²

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1
Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \(y = x^2\) to rewrite it in terms of \(r\) and \(\theta\).
This substitution gives \(r \sin{\theta} = (r \cos{\theta})^2\).
Simplify the right side to get \(r \sin{\theta} = r^2 \cos^2{\theta}\).
To isolate \(r\), divide both sides by \(r\) (assuming \(r \neq 0\)), resulting in \(\sin{\theta} = r \cos^2{\theta}\). Then solve for \(r\) to express the equation fully in polar form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding the relationship between these systems is essential for converting equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Cartesian and Polar Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). To convert an equation from Cartesian to polar form, substitute x and y with these expressions and simplify to express the equation in terms of r and θ.
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Intro to Polar Coordinates

Manipulating and Simplifying Trigonometric Expressions

After substitution, the resulting equation often involves trigonometric functions like sine and cosine. Being able to manipulate and simplify these expressions is crucial to rewrite the equation clearly in polar form, such as isolating r or expressing it as a function of θ.
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Simplifying Trig Expressions
Related Practice
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

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = sin 3θ

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

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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