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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.3.36
Capitolo 11, Problema 11.3.36

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r² = 4r sin θ

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1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r^2 = 4r \sin \theta\).
Divide both sides of the equation by \(r\) (noting that \(r \neq 0\)) to simplify: \(r = 4 \sin \theta\).
Substitute \(r = \sqrt{x^2 + y^2}\) and \(\sin \theta = \frac{y}{r}\) into the simplified equation: \(\sqrt{x^2 + y^2} = 4 \cdot \frac{y}{\sqrt{x^2 + y^2}}\).
Multiply both sides by \(\sqrt{x^2 + y^2}\) to eliminate the denominator, then simplify and rearrange the resulting equation to express it purely in terms of \(x\) and \(y\). This will give the Cartesian form of the curve.

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Polar to Cartesian Coordinate Conversion

Polar coordinates (r, θ) relate to Cartesian coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Understanding these relationships allows one to rewrite polar equations in terms of x and y, facilitating analysis using Cartesian methods.
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05:32
Intro to Polar Coordinates

Algebraic Manipulation of Equations

After substituting polar expressions with Cartesian equivalents, algebraic skills are essential to simplify and rearrange the equation. This process helps in identifying the standard form of the curve and makes it easier to classify the graph.
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08:02
Parameterizing Equations

Graph Identification from Equations

Recognizing the type of graph from its Cartesian equation involves knowledge of common curves like circles, lines, parabolas, and ellipses. Identifying these shapes helps in visualizing and describing the graph represented by the original polar equation.
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04:47
Introduction to Parametric Equations
Pratica correlata
Domanda del libro di testo

Lengths of Curves


Find the lengths of the curves in Exercises 25–30.


x = cos t, y = t + sin t, 0 ≤ t ≤ π

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Examples of Polar Equations


[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.


r = −2 cos θ

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Domanda del libro di testo

Cartesian to Polar Equations


Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.


x - y = 3

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Domanda del libro di testo

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Domanda del libro di testo

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the line segment with endpoints (-1,3) and (3,-2)

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Finding Cartesian from Parametric Equations


Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.


x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

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