Textbook Question
51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.
The limaçon r=3−6cosθ
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 12.2.77a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).
Verified step by step guidance1
Recall the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\) (adjusted for the correct quadrant).
Calculate the radius \(r\) for the point \((-2, 2)\): \(r = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\).
Determine the angle \(\theta\): Since \(x = -2\) and \(y = 2\), the point lies in the second quadrant. The reference angle is \(\arctan\left(\frac{2}{-2}\right) = \arctan(-1)\), but we must add \(\pi\) to get the correct angle in the second quadrant, so \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).
Check each given polar coordinate to see if it corresponds to the Cartesian point \((-2, 2)\) by converting back to Cartesian using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\):
- For \((2\sqrt{2}, \frac{3\pi}{4})\), verify if \(x = 2\sqrt{2} \cos(\frac{3\pi}{4})\) and \(y = 2\sqrt{2} \sin(\frac{3\pi}{4})\) equal \((-2, 2)\).
- Repeat this for \((2\sqrt{2}, \frac{11\pi}{4})\), \((2\sqrt{2}, -\frac{5\pi}{4})\), and \((-2\sqrt{2}, -\frac{\pi}{4})\).
Remember that polar coordinates are not unique: adding or subtracting multiples of \(2\pi\) to \(\theta\) gives the same point, and using a negative \(r\) with \(\theta\) shifted by \(\pi\) also represents the same point. Use these facts to explain which of the given coordinates are valid representations of \((-2, 2)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conversion between Cartesian and Polar Coordinates
This concept involves translating a point from Cartesian coordinates (x, y) to polar coordinates (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. The formulas are r = √(x² + y²) and θ = arctan(y/x), adjusted for the correct quadrant.
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Intro to Polar Coordinates
Multiple Representations of Polar Coordinates
A single point can have infinitely many polar coordinates because adding or subtracting multiples of 2π to the angle θ results in the same direction. Also, negative radius values with adjusted angles can represent the same point, reflecting the non-uniqueness of polar coordinates.
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05:32Intro to Polar Coordinates
Quadrant Considerations in Angle Determination
When converting to polar form, the angle θ must be placed in the correct quadrant based on the signs of x and y. This ensures the angle accurately represents the point's direction, as arctan alone only gives values between -π/2 and π/2.
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5:30Trig Values in Quadrants II, III, & IV
Related Practice
Textbook Question
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²
Textbook Question
Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.
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Textbook Question
67–72. Derivatives Consider the following parametric curves.
a. Determine dy/dx in terms of t and evaluate it at the given value of t.
x = cos t, y = 8 sin t; t = π/2
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Textbook Question
67–72. Derivatives Consider the following parametric curves.
a. Determine dy/dx in terms of t and evaluate it at the given value of t.
x = t + 1/t, y = t − 1/t; t = 1
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Textbook Question
(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933
a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.
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