Skip to main content
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

Verified step by step guidance
1
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 \( t \in [0, 2\pi] \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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.
Recommended video:
Guided course
08:02
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.
Recommended video:
Guided course
06:03
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.
Recommended video:
Guided course
05:59
Eliminating the Parameter
Related Practice
Textbook Question

Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π

87
views
Textbook Question

{Use of Tech} Implicit function graph Explain and carry out a method for graphing the curve x = 1 + cos² y − sin² y using parametric equations and a graphing utility.

24
views
Textbook Question

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


r = 1 - cos θ

67
views
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)

64
views
Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π

61
views
Textbook Question

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


The complete cardioid r = 4 + 4 sin θ

57
views