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 5.1.53

Chapter 5, Problem 5.1.53

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 − 3i)(1 − i) − (3 − i)(3 + i)

Verified step by step guidance

Recall that to multiply two complex numbers, use the distributive property (FOIL method): for \((a + bi)(c + di)\), the product is \((ac - bd) + (ad + bc)i\).

First, multiply the complex numbers in the first product: \((2 - 3i)(1 - i)\). Apply the distributive property carefully.

Next, multiply the complex numbers in the second product: \((3 - i)(3 + i)\). Remember that \((a - bi)(a + bi) = a^2 + b^2\) because it is a difference of squares.

After finding both products, subtract the second product from the first as indicated: \((2 - 3i)(1 - i) - (3 - i)(3 + i)\).

Finally, combine like terms (real parts together and imaginary parts together) to write the result 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 Number Multiplication

Multiplying complex numbers involves using the distributive property (FOIL) and applying the rule i² = -1. Each term in the first complex number is multiplied by each term in the second, then like terms are combined to simplify the expression.

Complex Number Subtraction

Subtracting complex numbers requires subtracting their real parts and imaginary parts separately. This operation is straightforward once both complex numbers are expressed in the form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Writing the result in this form means simplifying the expression so that it clearly shows the real and imaginary components.

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Related Practice

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

x² + y² = 16

Textbook Question

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

Textbook Question

In Exercises 1–10, perform the indicated operations and write the result in standard form. (7 + 8i)(7 − 8i)

Textbook Question

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−√3,−1)

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x² + y² = 9

Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 10° + i sin 10°)]³

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