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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 12.2.52
Chapter 12, Problem 12.2.52

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


(x - 1)² + y² = 1

Verified step by step guidance
1
Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \((x - 1)^2 + y^2 = 1\) to rewrite it in terms of \(r\) and \(\theta\).
Expand the expression: \((r \cos{\theta} - 1)^2 + (r \sin{\theta})^2 = 1\).
Simplify the equation by expanding the square and combining like terms: \((r^2 \cos^2{\theta} - 2r \cos{\theta} + 1) + r^2 \sin^2{\theta} = 1\).
Use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta} = 1\) to combine terms and then isolate \(r\) to express the equation purely in terms of \(r\) and \(\theta\).

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Key Concepts

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

Polar Coordinates System

Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. This system is useful for describing curves that are circular or have rotational symmetry.
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Intro to Polar Coordinates

Conversion Formulas Between Cartesian and Polar Coordinates

To convert from Cartesian (x, y) to polar (r, θ), use x = r cos θ and y = r sin θ. Conversely, r = √(x² + y²) and θ = arctan(y/x). These formulas allow rewriting equations from Cartesian form into polar form.
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Equation of a Circle in Polar Coordinates

A circle centered at (a, 0) with radius R in Cartesian coordinates can be expressed in polar form by substituting x = r cos θ and y = r sin θ into the circle's equation. This often results in an equation involving r and θ that describes the same circle in polar terms.
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Related Practice
Textbook Question

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=cos t+t sin t,y=sin t−t cos t; t=π/4

Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 4 + sin θ; (4, 0) and (3, 3π/2)

Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.


Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

Textbook Question

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.

Textbook Question

Spiral arc length Consider the spiral r=4θ, for θ≥0.


a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.