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

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

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

Problem 41

Chapter 5, Problem 41

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

Verified step by step guidance

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 \(a + bi\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

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Related Practice

Textbook Question

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)

Textbook Question

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

Textbook Question

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

Textbook Question

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

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Textbook Question

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

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Textbook Question

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