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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 15
Chapter 5, Problem 15

In Exercises 9–20, find each product and write the result in standard form.


(3 + 5i)(3 − 5i)

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1
Recognize that the expression is a product of two complex conjugates: \((3 + 5i)\) and \((3 - 5i)\).
Recall the formula for the product of conjugates: \((a + bi)(a - bi) = a^2 + b^2\), where \(a\) and \(b\) are real numbers.
Identify \(a = 3\) and \(b = 5\) from the given expression.
Calculate \(a^2\) and \(b^2\) separately: \$3^2\( and \)5^2$.
Add the results from the previous step to write the product in standard form: \(a^2 + b^2\).

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Key Concepts

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

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing a complex number in standard form means expressing it explicitly as a sum of its real and imaginary components.
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Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method), multiplying each term in the first complex number by each term in the second. Remember that i² equals -1, which simplifies the product.
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Difference of Squares Formula

The product (a + b)(a - b) equals a² - b². This formula applies to complex conjugates like (3 + 5i)(3 - 5i), simplifying the multiplication by turning it into a difference of squares involving real numbers and imaginary parts.
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Verifying Identities with Sum and Difference Formulas
Related Practice
Textbook Question
In Exercises 13–34, test for symmetry and then graph each polar equation.r = 2 + 2 cos θ
Textbook Question

Eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = √t , y = t + 1; −∞ < t < ∞

Textbook Question

In Exercises 9–20, find each product and write the result in standard form. (−5 + i)(−5 − i)

Textbook Question

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (3, 4π/3)

Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

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Textbook Question

Write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 60° + i sin 60°)

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