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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 12.2.16
Chapter 12, Problem 12.2.16

15–22. Sets in polar coordinates Sketch the following sets of points.


r = 3

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1
Understand that the equation \(r = 3\) in polar coordinates represents all points that are at a distance of 3 units from the origin (pole).
Recall that in polar coordinates, \(r\) is the radius or distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.
Since \(r\) is constant and equal to 3, this means the set of points forms a circle centered at the origin with radius 3.
To sketch this, draw a circle with center at the origin (0,0) and radius 3 units on the polar coordinate plane.
Label the circle and note that it includes all points where \(r = 3\) for any angle \(\theta\) between \(0\) and \(2\pi\).

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Key Concepts

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

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. This system is especially useful for describing curves and regions with circular symmetry.
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Intro to Polar Coordinates

Graphing r = Constant in Polar Coordinates

The equation r = 3 describes all points that are exactly 3 units away from the origin, regardless of the angle θ. Graphically, this set forms a circle centered at the origin with radius 3. Understanding this helps in sketching curves defined by constant radius values.
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Intro to Polar Coordinates

Conversion Between Polar and Cartesian Coordinates

To better visualize or analyze polar equations, converting between polar (r, θ) and Cartesian (x, y) coordinates is useful. The formulas x = r cos θ and y = r sin θ allow translation of points, enabling the use of familiar Cartesian graphing techniques for polar curves.
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Intro to Polar Coordinates
Related Practice
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

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola that opens to the right with directrix x = -4

Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 


A hyperbola with vertices (±4, 0) and foci (±6, 0)

Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

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

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.