Textbook Question
Polar Coordinates
Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.
a. (2, π/2)
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.3.9c
Cartesian to Polar Coordinates
Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.
c. (−1, √3)
Verified step by step guidance1
Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\):
\(r = \sqrt{x^2 + y^2}\)
\(\theta = \arctan\left(\frac{y}{x}\right)\), but be careful to determine the correct quadrant for \(\theta\).
Substitute the given point \((-1, \sqrt{3})\) into the formula for \(r\):
\(r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3}\).
Calculate \(r\) from the expression above (do not finalize the numeric value here, just set up the expression).
Find the angle \(\theta\) by substituting \(x = -1\) and \(y = \sqrt{3}\) into \(\theta = \arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{\sqrt{3}}{-1}\right)\).
Determine the correct quadrant for \(\theta\) since \(x\) is negative and \(y\) is positive, which places the point in the second quadrant. Adjust \(\theta\) accordingly to ensure \(0 \leq \theta \leq 2\pi\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conversion between Cartesian and Polar Coordinates
This concept involves translating a point from the Cartesian coordinate system (x, y) to the polar coordinate system (r, θ). The radius r is the distance from the origin to the point, calculated as r = √(x² + y²), and the angle θ is the counterclockwise angle from the positive x-axis, found using θ = arctan(y/x), adjusted for the correct quadrant.
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Intro to Polar Coordinates
Determining the Correct Angle θ in Polar Coordinates
Since arctan(y/x) only returns values between -π/2 and π/2, it is essential to consider the signs of x and y to place θ in the correct quadrant. For points in different quadrants, θ may need to be adjusted by adding π or 2π to ensure it lies within the interval 0 ≤ θ ≤ 2π.
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05:32Intro to Polar Coordinates
Interpretation of Negative Radius r in Polar Coordinates
While r is typically non-negative, polar coordinates allow r to be negative, which means the point is located in the direction opposite to the angle θ. If r ≤ 0, the angle θ is adjusted by adding π to point in the opposite direction, effectively representing the same point with a negative radius.
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Related Practice
Textbook Question
Polar to Cartesian Coordinates
Find the Cartesian coordinates of the following points, given in polar coordinates.
c. (0, π/2)
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Textbook Question
Theory and Examples
Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x² + 4y² = 36 about the y−axis.
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Textbook Question
Cartesian to Polar Coordinates
Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.
b. (-3,0)
1
views