Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 67

Capitolo 5, Problema 67

In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + cos θ

Guida verificata passo dopo passo

Identify the type of polar equation: The given equation \( r = 2 + \cos \theta \) is a limaçon, which is a type of polar curve.

Test for symmetry: Check for symmetry with respect to the polar axis (x-axis), the line \( \theta = \frac{\pi}{2} \) (y-axis), and the pole (origin). For this equation, substitute \( \theta \) with \( -\theta \) to test for x-axis symmetry, and substitute \( r \) with \(-r\) to test for origin symmetry.

Determine key points: Calculate \( r \) for key angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) to find specific points on the graph.

Plot the points: Use the calculated points to plot the graph on polar coordinates. Remember that the graph will be symmetric based on the tests conducted earlier.

Sketch the curve: Connect the plotted points smoothly to form the limaçon shape, ensuring to reflect any symmetries identified in the earlier steps.

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:

Concetti chiave

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

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In the equation r = 2 + cos θ, 'r' denotes the radius or distance from the origin, while 'θ' is the angle measured from the positive x-axis. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Video consigliato:

Percorso guidato

05:32

Intro to Polar Coordinates

Symmetry in Polar Graphs

Symmetry in polar graphs can be determined by analyzing the equation with respect to specific angles. A polar graph is symmetric about the polar axis if replacing θ with -θ yields the same equation, and it is symmetric about the line θ = π/2 if replacing θ with π - θ does so. Testing for symmetry helps in sketching the graph accurately and understanding its properties.

Video consigliato:

Percorso guidato

3:19

Cardioids

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of 'r' for various angles 'θ'. The equation r = 2 + cos θ describes a limaçon shape, which can exhibit different characteristics based on the coefficients involved. To graph it effectively, one must calculate 'r' for key angles (like 0, π/2, π, and 3π/2) and then plot these points in the polar coordinate system.

Video consigliato:

Percorso guidato

3:47

Introduction to Common Polar Equations

Pratica correlata

Domanda del libro di testo

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 27(cos 306° + i sin 306°)

512

views

Domanda del libro di testo

In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fourth roots of 81 (cos 4π/3 + i sin 4π/3)

530

views

Domanda del libro di testo

In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ −8 + √−32 / 24

731

views

Domanda del libro di testo

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 8(cos 210° + i sin 210°)

610

views

Domanda del libro di testo

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 12 cos θ

973

views

Domanda del libro di testo

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

925

views