Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.3.1

Chapter 12, Problem 12.3.1

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

Verified step by step guidance

Recall the relationship between polar and Cartesian coordinates: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r\) is the radius and \(\theta\) is the angle parameter.

Given the polar equation \(r = f(\theta)\), substitute \(r\) into the Cartesian coordinate formulas to express \(x\) and \(y\) in terms of \(\theta\).

Write the parametric equations as \(x(\theta) = f(\theta) \cdot \cos(\theta)\) and \(y(\theta) = f(\theta) \cdot \sin(\theta)\).

Note that \(\theta\) serves as the parameter that varies, typically within an interval such as \([0, 2\pi]\), to trace the curve in the Cartesian plane.

Thus, the polar equation \(r = f(\theta)\) is expressed in parametric form as \(\left(x(\theta), y(\theta)\right) = \left(f(\theta) \cos(\theta), f(\theta) \sin(\theta)\right)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle (r, θ) from the origin. The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential to convert between polar and Cartesian forms.

Parametric Equations

Parametric equations express coordinates as functions of a parameter, often denoted t or θ. Instead of y as a function of x, both x and y are defined separately in terms of the parameter, allowing more flexible representations of curves, including those defined in polar form.

Conversion from Polar to Cartesian Coordinates

To convert polar equations to Cartesian form, use the relationships x = r cos(θ) and y = r sin(θ). When r is given as a function of θ, substituting r = f(θ) into these formulas yields parametric equations x(θ) and y(θ) in Cartesian coordinates.

Recommended video:

05:32

Intro to Polar Coordinates

Related Practice

Textbook Question

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

The region common to the circles r = 2 sin θ and r = 1

views

Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(1, √3)

views

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.

The upper half of the parabola x=y ², originating at (0, 0)

views

Textbook Question

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

The region inside the lemniscate r² = 6 sin 2θ

views

Textbook Question

75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

views

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 = 2 - 2 sin θ b

views