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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 12.1.31
Capitolo 12, Problema 12.1.31

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=2 sin 8t, y=2 cos 8t 

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Start with the given parametric equations: \(x = 2 \sin(8t)\) and \(y = 2 \cos(8t)\).
Isolate the trigonometric functions by dividing both \(x\) and \(y\) by 2: \(\frac{x}{2} = \sin(8t)\) and \(\frac{y}{2} = \cos(8t)\).
Recall the Pythagorean identity for sine and cosine: \(\sin^2(\theta) + \cos^2(\theta) = 1\) for any angle \(\theta\).
Substitute \(\sin(8t)\) and \(\cos(8t)\) with \(\frac{x}{2}\) and \(\frac{y}{2}\) respectively in the identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1\).
Simplify the equation to get a single equation in terms of \(x\) and \(y\): \(\frac{x^2}{4} + \frac{y^2}{4} = 1\).

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Parametric Equations

Parametric equations express the coordinates x and y as functions of a third variable, usually t, called the parameter. Understanding how x and y depend on t allows us to describe curves that may not be functions in the traditional sense.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the variable t, resulting in a single equation relating x and y. This often requires using trigonometric identities or algebraic techniques to combine the expressions.
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Eliminating the Parameter

Trigonometric Identities

Trigonometric identities, such as sin²θ + cos²θ = 1, are essential tools for eliminating parameters when x and y are defined using sine and cosine functions. Applying these identities helps convert parametric forms into standard Cartesian equations.
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Verifying Trig Equations as Identities
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Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

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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.

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39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

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15–22. Sets in polar coordinates Sketch the following sets of points.


2 ≤ r ≤ 8

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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 = 12y

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37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = 6 cos θ + 8 sin θ

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