Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.2.1

Chapter 12, Problem 12.2.1

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.

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Recall that a point in polar coordinates is given by \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.

For the point \((2, \frac{\pi}{6})\), find alternative coordinates by adding \(2\pi\) to the angle or by using the negative radius with an adjusted angle. Specifically, one alternative is \((2, \frac{\pi}{6} + 2\pi)\), and another is \((-2, \frac{\pi}{6} + \pi)\).

For the point \((-3, -\frac{\pi}{2})\), similarly find alternatives by adding \(2\pi\) to the angle or by changing the sign of the radius and adjusting the angle by \(\pi\). One alternative is \((-3, -\frac{\pi}{2} + 2\pi)\), and another is \((3, -\frac{\pi}{2} + \pi)\).

Write down the two alternative coordinate pairs for each point explicitly, ensuring the angles are within a standard range (usually between \(0\) and \(2\pi\) or \(-\pi\) and \(\pi\)) if desired.

Verify that all coordinate pairs represent the same points by converting them to Cartesian coordinates using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) if needed.

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Key Concepts

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

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle, written as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how to interpret these coordinates is essential for plotting points correctly.

Equivalent Polar Coordinates

A single point in the plane can have multiple polar coordinate representations. This occurs because adding or subtracting 2π to the angle θ or changing the sign of r while adjusting θ by π results in the same point. Recognizing these equivalences helps in finding alternative coordinate pairs.

Plotting Points with Negative Radius or Angle

When the radius r is negative, the point is plotted in the direction opposite to the angle θ. Similarly, negative angles are measured clockwise from the positive x-axis. Understanding these conventions is crucial for accurately locating points with negative values in polar coordinates.

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Intro to Polar Coordinates Example 1

Related Practice

Textbook Question

45–60. Areas of regions Find the area of the following regions.

The region common to the circles r = 2 sin θ and r = 1

Textbook Question

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

(1, √3)

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Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The upper half of the parabola x=y ², originating at (0, 0)

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Textbook Question

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(2, 7π/4)

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Textbook Question

45–60. Areas of regions Find the area of the following regions.

The region inside the lemniscate r² = 6 sin 2θ

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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