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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 12.1.19
Chapter 12, Problem 12.1.19

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 = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π

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1
Identify the given parametric equations: \(x = 3 \cos t\) and \(y = 3 \sin t\), with the parameter \(t\) ranging from \(\pi\) to \(2\pi\).
To eliminate the parameter \(t\), use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\). Express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\): \(\cos t = \frac{x}{3}\) and \(\sin t = \frac{y}{3}\).
Substitute these into the identity to get an equation involving only \(x\) and \(y\): \(\left(\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\).
Simplify the equation to obtain the Cartesian form: \(\frac{x^2}{9} + \frac{y^2}{9} = 1\), which can be rewritten as \(x^2 + y^2 = 9\).
For the description of the curve, recognize that \(x^2 + y^2 = 9\) represents a circle of radius 3 centered at the origin. Since \(t\) ranges from \(\pi\) to \(2\pi\), the curve corresponds to the lower half of the circle, traced from the point \((-3, 0)\) to \((3, 0)\) moving in the direction of increasing \(t\) (which is clockwise).

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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, often denoted as 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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Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to form a single equation relating x and y directly. This often requires using trigonometric identities or algebraic techniques to remove t and describe the curve in Cartesian form.
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Curve Orientation and Domain of Parameter

The orientation of a parametric curve is determined by the direction in which the parameter t increases. The given domain of t (π ≤ t ≤ 2π) specifies which portion of the curve is traced and the direction of traversal, important for understanding the curve's positive orientation.
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Related Practice
Textbook Question

Given three polar coordinate representations for the origin.

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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 the limaçon r = 2 + cos 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 = 3 + t, y = 1 − t; 0 ≤ t ≤ 1

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Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.


2 ≤ r ≤ 8

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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.


x² = 12y

Textbook Question

63–74. Arc length of polar curves Find the length of the following polar curves.


The complete circle r = a sin θ, where a > 0