Skip to main content
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 41
Capitolo 5, Problema 41

In Exercises 37–44, find the product of the complex numbers. Leave answers in polar form. z₁ = cos π/4 + i sin π/4 z₂ = cos π/3 + i sin π/3

Guida verificata passo dopo passo
1
Recall that complex numbers in the form \(z = \cos \theta + i \sin \theta\) are expressed in polar form, where \(\theta\) is the argument (angle) of the complex number.
To find the product of two complex numbers in polar form, use the formula: if \(z_1 = \cos \alpha + i \sin \alpha\) and \(z_2 = \cos \beta + i \sin \beta\), then their product is \(z_1 z_2 = \cos(\alpha + \beta) + i \sin(\alpha + \beta)\).
Identify the angles for the given complex numbers: \(\alpha = \frac{\pi}{4}\) for \(z_1\) and \(\beta = \frac{\pi}{3}\) for \(z_2\).
Add the angles: calculate \(\alpha + \beta = \frac{\pi}{4} + \frac{\pi}{3}\), which will be the argument of the product.
Write the product in polar form as \(z_1 z_2 = \cos\left(\frac{\pi}{4} + \frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{4} + \frac{\pi}{3}\right)\).

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:
4m

Concetti chiave

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

Complex Numbers in Polar Form

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form highlights the geometric interpretation of complex numbers as points or vectors in the plane, making multiplication and division more intuitive.
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)]. This simplifies the product calculation significantly.
Video consigliato:
Percorso guidato
04:47
Complex Numbers In Polar Form

Euler's Formula and Trigonometric Representation

Euler's formula states that e^(iθ) = cos θ + i sin θ, linking exponential and trigonometric forms of complex numbers. This connection helps understand the polar form and the behavior of complex multiplication as rotation and scaling in the complex plane.
Video consigliato:
Percorso guidato
6:36
Quadratic Formula
Pratica correlata
Domanda del libro di testo

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 1 / 1−cos θ

736
views
Domanda del libro di testo

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−2, 2)

794
views
Domanda del libro di testo

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _ (2,−2√3)

720
views
Domanda del libro di testo

In Exercises 37–52, perform the indicated operations and write the result in standard form. __ (−2 + √−4)²

648
views
Domanda del libro di testo

In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___ 5√−16 + 3√−81

678
views
Domanda del libro di testo

In Exercises 37–52, perform the indicated operations and write the result in standard form. __ (−3 − √−7)²

698
views