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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.R.36a
Capitolo 12, Problema 12.R.36a

Polar conversion Consider the equation r=4/(sinθ+cosθ). 


a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

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1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos\theta\), \(y = r \sin\theta\), and \(r = \sqrt{x^2 + y^2}\).
Start with the given polar equation: \(r = \frac{4}{\sin\theta + \cos\theta}\).
Multiply both sides of the equation by \((\sin\theta + \cos\theta)\) to get: \(r (\sin\theta + \cos\theta) = 4\).
Substitute \(r \sin\theta = y\) and \(r \cos\theta = x\) into the equation, yielding \(y + x = 4\).
Recognize that the equation \(x + y = 4\) is a linear equation representing a straight line in Cartesian coordinates.

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Polar to Cartesian Coordinate Conversion

This involves translating equations from polar form (r, θ) to Cartesian form (x, y) using the relationships x = r cosθ and y = r sinθ. Understanding these conversions allows one to rewrite polar equations in terms of x and y, facilitating analysis using familiar Cartesian methods.
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Intro to Polar Coordinates

Trigonometric Identities and Manipulation

Trigonometric identities, such as expressing sinθ and cosθ in terms of x and y, or using sum formulas, are essential for simplifying and rearranging equations during conversion. Mastery of these identities helps in isolating variables and recognizing standard curve forms.
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Verifying Trig Equations as Identities

Identification of Conic Sections

After converting to Cartesian form, recognizing the resulting equation as a conic section (circle, ellipse, parabola, or hyperbola) is crucial. This involves comparing the equation to standard forms and understanding geometric properties to classify the curve accurately.
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Percorso guidato
5:33
Parabolas as Conic Sections
Pratica correlata
Domanda del libro di testo

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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Domanda del libro di testo

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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Domanda del libro di testo

27–32. Polar curves Graph the following equations.


r = 3 sin 4θ

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Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

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Domanda del libro di testo

7–8. Parametric curves and tangent lines

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

x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

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Domanda del libro di testo

Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?

b. r=(½)+sinθ

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