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 12.3.5
Capitolo 12, Problema 12.3.5

What is the slope of the line θ=π/3?

Guida verificata passo dopo passo
1
Recognize that the equation \( \theta = \frac{\pi}{3} \) represents a line in polar coordinates where the angle \( \theta \) is constant at \( \frac{\pi}{3} \).
Recall that in polar coordinates, \( \theta \) is the angle measured from the positive x-axis, so the line \( \theta = \frac{\pi}{3} \) is a straight line passing through the origin making an angle of \( \frac{\pi}{3} \) with the x-axis.
To find the slope of this line in Cartesian coordinates, use the relationship between slope \( m \) and angle \( \theta \): \[ m = \tan(\theta) \].
Substitute \( \theta = \frac{\pi}{3} \) into the formula to express the slope as \( m = \tan\left(\frac{\pi}{3}\right) \).
This expression gives the slope of the line \( \theta = \frac{\pi}{3} \) in Cartesian coordinates.

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:
1m

Concetti chiave

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

Polar Coordinates and Lines

In polar coordinates, a line defined by θ = constant represents all points making a fixed angle θ with the positive x-axis. This line passes through the origin and extends outward at that angle, differing from the Cartesian form y = mx + b.
Video consigliato:
05:32
Intro to Polar Coordinates

Relationship Between Angle and Slope

The slope of a line in Cartesian coordinates is the tangent of the angle it makes with the positive x-axis. Thus, for a line at angle θ, the slope m = tan(θ), linking angular direction to the line's steepness.
Video consigliato:
05:45
Understanding Slope Fields

Calculating Slope from Given Angle

To find the slope of the line θ = π/3, compute m = tan(π/3). Since tan(π/3) = √3, the slope of the line is √3, indicating a steep positive incline in the Cartesian plane.
Video consigliato:
03:51
Understanding Slope Fields Example 3
Pratica correlata
Domanda del libro di testo

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


25y² - 4x² = 100

22
views
Domanda del libro di testo

Cartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r² = a² cos 2θ, where a is a real number.

73
views
Domanda del libro di testo

What is the polar equation of the vertical line x = 5?

70
views
Domanda del libro di testo

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

109
views
Domanda del libro di testo

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 line segment starting at P(0, 0) and ending at Q(2, 8)

79
views
Domanda del libro di testo

53–56. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.


An ellipse with vertices (0, ±9) and eccentricity ¼ 

31
views