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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.3.9c
Capitolo 11, Problema 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)

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Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\): \(r = \sqrt{x^2 + y^2}\) \(\theta = \arctan\left(\frac{y}{x}\right)\), 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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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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Intro 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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Intro to Polar Coordinates
Pratica correlata
Domanda del libro di testo

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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Domanda del libro di testo

Polar to Cartesian Coordinates


Find the Cartesian coordinates of the following points, given in polar coordinates.


c. (0, π/2)

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Domanda del libro di testo

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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Domanda del libro di testo

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)

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