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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 12.1.21
Chapter 12, Problem 12.1.21

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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Start with the given parametric equations: \(x = \cos t\) and \(y = \sin^{2} t\), where \(0 \leq t \leq \pi\).
Recall the Pythagorean identity: \(\sin^{2} t + \cos^{2} t = 1\). Use this to express \(\sin^{2} t\) in terms of \(\cos t\).
Since \(y = \sin^{2} t\), rewrite it as \(y = 1 - \cos^{2} t\). Substitute \(x = \cos t\) into this to eliminate the parameter \(t\).
This substitution gives the Cartesian equation relating \(x\) and \(y\): \(y = 1 - x^{2}\).
To describe the curve, recognize that \(y = 1 - x^{2}\) is a downward-opening parabola. The parameter range \(0 \leq t \leq \pi\) corresponds to \(x\) values from \(1\) to \(-1\). The positive orientation follows the direction of increasing \(t\), which moves from \(x=1\) to \(x=-1\) along the curve.

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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 and motions.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to find a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to remove t and express y explicitly or implicitly in terms of x.
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Eliminating the Parameter

Curve Orientation and Description

The orientation of a parametric curve indicates the direction in which the curve is traced as the parameter increases. Describing the curve involves identifying its shape (e.g., circle, ellipse) and specifying the interval of t to understand the portion and direction of the curve.
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Summary of Curve Sketching
Related Practice
Textbook Question

53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.


r = 1 - cos θ

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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

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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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

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


{Use of Tech} The complete limaçon r=4−2cosθ

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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