Domanda del libro di testo
In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)
841
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 37
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 37Capitolo 5, Problema 37
In Exercises 37–44, find the product of the complex numbers. Leave answers in polar form. z₁ = 6(cos 20° + i sin 20°) z₂ = 5(cos 50° + i sin 50°)
Guida verificata passo dopo passo1
Recall that when multiplying two complex numbers in polar form, the magnitudes multiply and the angles add. This is based on De Moivre's Theorem.
Identify the magnitudes and angles of the given complex numbers: For \(z_1 = 6(\cos 20^\circ + i \sin 20^\circ)\), the magnitude is 6 and the angle is \(20^\circ\). For \(z_2 = 5(\cos 50^\circ + i \sin 50^\circ)\), the magnitude is 5 and the angle is \(50^\circ\).
Multiply the magnitudes: \(r = 6 \times 5\).
Add the angles: \(\theta = 20^\circ + 50^\circ\).
Write the product in polar form as \(r(\cos \theta + i \sin \theta)\) using the values found in steps 3 and 4.
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:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Polar Form of Complex Numbers
A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude (modulus) and θ is the argument (angle). This form is useful for multiplication and division because it separates the magnitude and angle components.
Video consigliato:
Percorso guidato
04:47
Complex Numbers In Polar Form
Multiplication of Complex Numbers in Polar Form
When multiplying two complex numbers in polar form, multiply their magnitudes and add their angles. Specifically, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂ [cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)].
Video consigliato:
Percorso guidato
04:47Complex Numbers In Polar Form
De Moivre's Theorem
De Moivre's theorem states that raising a complex number in polar form to a power n results in rⁿ [cos(nθ) + i sin(nθ)]. While not directly required here, it underpins the manipulation of angles and magnitudes in polar form and helps understand multiplication as repeated addition of angles.
Video consigliato:
Percorso guidato
03:41Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)
Pratica correlata
Domanda del libro di testo
In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (−4, π/2)
893
views
Domanda del libro di testo
In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___ √−64 − √−25
690
views
Domanda del libro di testo
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.
x = 5 sec t, y = 3 tan t
519
views
Domanda del libro di testo
In Exercises 29–36, simplify and write the result in standard form. ____________ √1² − 4 ⋅ 0.5 ⋅ 5
646
views
Domanda del libro di testo
In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fifth roots of −1 − i
574
views