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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 12.2.33
Chapter 12, Problem 12.2.33

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


(1, √3)

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Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\): \(r = \sqrt{x^2 + y^2}\) \(\theta = \arctan\left(\frac{y}{x}\right)\)
Calculate the radius \(r\) by substituting \(x = 1\) and \(y = \sqrt{3}\) into the formula: \(r = \sqrt{1^2 + (\sqrt{3})^2}\)
Calculate the angle \(\theta\) using the arctangent formula: \(\theta = \arctan\left(\frac{\sqrt{3}}{1}\right)\)
Express the polar coordinates as \((r, \theta)\) using the values found in steps 2 and 3.
For the second way, express the angle \(\theta\) in degrees instead of radians or use a known angle value (e.g., \(\theta = \frac{\pi}{3}\) radians or \(60^\circ\)) to write the polar coordinates as \((r, \theta)\).

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Key Concepts

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

Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates describe points by their distance from the origin (r) and the angle (θ) from the positive x-axis. Understanding both systems is essential for converting between them.
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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 r = √(x² + y²) to find the radius and θ = arctan(y/x) to find the angle. Adjust θ based on the quadrant to get the correct direction.
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Intro to Polar Coordinates

Multiple Representations of Polar Coordinates

Polar coordinates are not unique; the same point can be represented by adding multiples of 2π to θ or by using negative r values with adjusted angles. Recognizing these alternatives allows expressing coordinates in at least two different valid ways.
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Intro to Polar Coordinates
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

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

What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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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

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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