Textbook Question
53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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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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:
05:32
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.
Recommended video:
7:17Verifying 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.
Recommended video:
5:33Parabolas as Conic Sections
Related Practice
Textbook Question
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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Textbook Question
27–32. Polar curves Graph the following equations.
r = 3 sin 4θ
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Textbook Question
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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Textbook Question
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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Textbook Question
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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