Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.2.35
Capitolo 12, Problema 12.2.35

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


(-4, 4√3)

Guida verificata passo dopo passo
1
Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\): \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\), but be careful with the quadrant of the point when determining \(\theta\).
Calculate the radius \(r\) using the given coordinates \(x = -4\) and \(y = 4\sqrt{3}\): \(r = \sqrt{(-4)^2 + (4\sqrt{3})^2}\).
Find the angle \(\theta\) by computing \(\arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{4\sqrt{3}}{-4}\right)\), then adjust \(\theta\) to the correct quadrant since \(x\) is negative and \(y\) is positive (which places the point in the second quadrant).
Express the polar coordinates as \((r, \theta)\) where \(r\) is the radius found in step 2 and \(\theta\) is the angle found in step 3, typically in radians.
For the second method, use the fact that \(\theta\) can also be expressed as \(\pi - |\arctan(\frac{y}{|x|})|\) to directly find the angle in the second quadrant, then write the polar coordinates again as \((r, \theta)\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates describe points by a distance from the origin (r) and an angle (θ) from the positive x-axis. Understanding both systems is essential for converting between them.
Video consigliato:
05:32
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. Adjustments to θ may be needed depending on the quadrant where the point lies.
Video consigliato:
05:32
Intro to Polar Coordinates

Multiple Representations of Polar Coordinates

Polar coordinates are not unique; the same point can be represented by different pairs (r, θ) by adding or subtracting full rotations (2π) to the angle or by using negative radius values with adjusted angles. Recognizing these variations helps express coordinates in multiple valid ways.
Video consigliato:
05:32
Intro to Polar Coordinates
Pratica correlata
Domanda del libro di testo

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 is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?


47
views
Domanda del libro di testo

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 2 cos θ and r = 1 + cos θ

62
views
Domanda del libro di testo

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

59
views
Domanda del libro di testo

84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is

L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

68
views
Domanda del libro di testo

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


(x - 1)² + y² = 1

71
views
Domanda del libro di testo

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 2θ; (π/2, π/4)

88
views