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In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form. $z=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)$

   In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.
 $z=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 39

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Step 1

The polar form of a complex number is \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle). For \( z = 2\left(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\right) \), the magnitude \( r = 2 \) Show more…

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In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form. $z=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)$

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Key Concepts

Complex Numbers in Polar Form

This concept involves representing a complex number as r(cos ? + i sin ?), where r is the modulus reflecting the distance from the origin in the complex plane, and ? is the argument indicating the angle at which the number is located. This form is particularly useful for operations like multiplication and division because it directly utilizes the geometric properties of complex numbers.

Multiplication of Complex Numbers in Polar Form

In polar form, the multiplication of complex numbers is simplified by multiplying their moduli and adding their arguments. This rule leverages the properties of trigonometric functions and makes it easier to handle products compared to using the standard rectangular form.

Division of Complex Numbers in Polar Form

Dividing complex numbers in polar form involves dividing the moduli and subtracting the arguments. This method provides a straightforward way to perform the division operation, aligning with the geometric interpretation of complex numbers by focusing on their magnitude and angle.

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