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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.R.36a
Chapter 12, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
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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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Parabolas as Conic Sections
Related Practice
Textbook Question

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

r =3 − 6 cos θ

Textbook Question

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

Textbook Question

19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve

The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2

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Textbook Question

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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Textbook Question

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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Textbook Question

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

e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.