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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.R.11a
Capitolo 12, Problema 12.R.11a

10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)

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1
Recognize that the given parametric equations are \(x = 3\cos(-t)\) and \(y = 3\sin(-t) - 1\) with the parameter \(t\) in the interval \(0 \leq t \leq \pi\).
Recall the trigonometric identities for cosine and sine of negative angles: \(\cos(-t) = \cos t\) and \(\sin(-t) = -\sin t\). Use these to rewrite the parametric equations as \(x = 3\cos t\) and \(y = -3\sin t - 1\).
Isolate the trigonometric functions from the parametric equations: \(\cos t = \frac{x}{3}\) and \(\sin t = -\frac{y + 1}{3}\).
Use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to eliminate the parameter \(t\). Substitute the expressions for \(\sin t\) and \(\cos t\) into this identity:
\[\left(\frac{x}{3}\right)^2 + \left(-\frac{y + 1}{3}\right)^2 = 1.\]
Simplify the equation to obtain a relation purely in terms of \(x\) and \(y\), which represents the Cartesian equation of the curve without the parameter \(t\).

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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 like circles or ellipses.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.
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Eliminating the Parameter

Trigonometric Identities

Trigonometric identities, such as sin(-t) = -sin(t) and cos(-t) = cos(t), are essential for simplifying parametric equations involving sine and cosine. These identities help transform and combine expressions to eliminate the parameter and find the Cartesian equation.
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Verifying Trig Equations as Identities
Pratica correlata
Domanda del libro di testo

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (±4, 0) and directrices x = ±2

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10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

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14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The circle x ² + y ² =9, generated clockwise

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53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

x² - y²/2 = 1

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Domanda del libro di testo

Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

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61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.


r = 3/(1 - 2 cos θ)

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