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.

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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 \(\left(x(\theta), y(\theta)\right) = \left(f(\theta) \cos(\theta), f(\theta) \sin(\theta)\right)\).

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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.

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Related Practice

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.

x²/9 + y²/4 = 1

Textbook Question

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Textbook Question

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

Textbook Question

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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 = 8 + 2t, y = 1; −∞ < t < ∞