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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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Intro to Polar Coordinates

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

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 is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?


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Textbook Question

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 2 cos θ and r = 1 + cos θ

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Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(-4, 4√3)

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Textbook Question

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

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Textbook Question

23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.

A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.

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Textbook Question

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-1, -π/3)

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