Domanda del libro di testo
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 80° + i sin 80°)]³
640
views
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Tutti i libri di testo
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.3.49
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.3.49Capitolo 5, Problema 5.3.49
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7
Guida verificata passo dopo passo1
Recall the relationships between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given rectangular equation \(3x + y = 7\) to get \(3(r \cos{\theta}) + r \sin{\theta} = 7\).
Factor out \(r\) from the left side: \(r(3 \cos{\theta} + \sin{\theta}) = 7\).
Solve for \(r\) by dividing both sides by \((3 \cos{\theta} + \sin{\theta})\): \(r = \frac{7}{3 \cos{\theta} + \sin{\theta}}\).
This expression gives the polar form of the equation, expressing \(r\) explicitly in terms of \(\theta\).
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Rectangular and Polar Coordinate Systems
Rectangular coordinates represent points using (x, y) on a plane, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems relate is essential for converting equations between them.
Video consigliato:
Percorso guidato
05:32
Intro to Polar Coordinates
Conversion Formulas Between Coordinates
The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of rectangular variables with polar expressions, enabling the transformation of equations from rectangular to polar form.
Video consigliato:
Percorso guidato
05:32Intro to Polar Coordinates
Expressing r in Terms of θ
After substituting x and y with r cos(θ) and r sin(θ), the goal is to isolate r on one side of the equation. This often involves algebraic manipulation to express r explicitly as a function of θ, which is the standard form for polar equations.
Video consigliato:
Percorso guidato
6:36Simplifying Trig Expressions
Pratica correlata
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 = 4 csc θ
912
views
Domanda del libro di testo
In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)
921
views
Domanda del libro di testo
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/12 + i sin π/12)]⁶
606
views
Domanda del libro di testo
In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. ___ ___ 5√(−16) + 3√(−81)
715
views
Domanda del libro di testo
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Circle: Center: (3,5); Radius: 6
676
views