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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 12.1.37
Chapter 12, Problem 12.1.37

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


A circle centered at the origin with radius 4, generated counterclockwise

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Recall the standard parametric equations for a circle centered at the origin with radius \( r \): \( x = r \cos(t) \) and \( y = r \sin(t) \).
Since the radius is 4, substitute \( r = 4 \) into the equations to get \( x = 4 \cos(t) \) and \( y = 4 \sin(t) \).
The parameter \( t \) represents the angle in radians measured from the positive x-axis, and it controls the position on the circle.
To generate the circle counterclockwise starting from the point \( (4,0) \), let \( t \) vary over the interval \( [0, 2\pi] \).
Thus, the parametric equations are \( x = 4 \cos(t) \), \( y = 4 \sin(t) \), with \( t \in [0, 2\pi] \).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
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Parameterizing Equations

Equation of a Circle

A circle centered at the origin with radius r satisfies x² + y² = r². To represent this circle parametrically, trigonometric functions sine and cosine are used, since they naturally describe circular motion.
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Parameterizing Equations of Circles & Ellipses

Parameter Interval and Direction

The parameter interval defines the portion of the curve traced by the parametric equations. For a full circle traced counterclockwise, t typically ranges from 0 to 2π, with x = r cos t and y = r sin t ensuring the correct orientation.
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Related Practice
Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

Textbook Question

Circles in general Show that the polar equation

r² - 2r r₀ cos(θ - θ₀) = R² - r₀²

describes a circle of radius R whose center has polar coordinates (r₀, θ₀)

Textbook Question

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. 


r = 3/(1 - cos θ)

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Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = sin² t; 0 ≤ t ≤ π

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Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 1 - sin θ; (1/2, π/6)

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