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 41

Capitolo 5, Problema 41

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

Guida verificata passo dopo passo

Recognize that the expression involves a complex number: $-2 + \sqrt{-4}$. Since $\sqrt{-4} = 2i$, rewrite the expression as $(-2 + 2i)^2$.

Recall the formula for squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = -2$ and $b = 2i$.

Calculate each term separately: $a^2 = (-2)^2$, $2ab = 2 \times (-2) \times 2i$, and $b^2 = (2i)^2$.

Simplify each term: $(-2)^2 = 4$, $2 \times (-2) \times 2i = -8i$, and $(2i)^2 = 4i^2$. Remember that $i^2 = -1$, so $4i^2 = 4 \times (-1) = -4$.

Combine all terms: $4 - 8i - 4$. Then, simplify the real parts $4 - 4$ and write the expression in standard form $a + bi$.

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.

Complex Numbers and Imaginary Unit

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to interpret and manipulate expressions involving √-4 requires recognizing that √-4 = 2i.

Video consigliato:

Percorso guidato

3:31

Introduction to Complex Numbers

Operations with Complex Numbers

Performing operations like addition, multiplication, and exponentiation on complex numbers follows algebraic rules, treating i as a variable but applying i² = -1 to simplify. Squaring a complex number involves expanding the binomial and simplifying using these rules.

Video consigliato:

Percorso guidato

4:22

Dividing Complex Numbers

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations, the result should be simplified and expressed clearly in this form, separating the real and imaginary parts.

Video consigliato:

Percorso guidato

04:47

Complex Numbers In Polar Form

Pratica correlata

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 = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

772

views

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. ___ ___ 5√−16 + 3√−81

678

views

Domanda del libro di testo

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

545

views