Converting a Complex Number to Polar Form: A Step-by-Step Guide

Euler's formula is a powerful mathematical tool that allows us to represent complex numbers in a more intuitive form. This article will guide you through the process of converting a given complex number to its polar form, providing a detailed walk-through of each step.

Introduction to Complex Numbers and Polar Form

A complex number is a number of the form a bi, where a and b are real numbers, and i is the imaginary unit, defined as i ?1. The polar form of a complex number is given by r (cos(θ) i sin(θ)), where r is the magnitude and θ is the angle with respect to the real axis.

Problem Setup

We are given a complex number:

Z (i - 1) / (cos(π/3) i sin(π/3))

Our goal is to convert this into its polar form.

Step 1: Simplify the Denominator

First, we use Euler's formula, which states:

e^iθ cos(θ) i sin(θ)

Applying this to the denominator:

cos(π/3) i sin(π/3) e^(iπ/3)

Thus, the complex number Z can be rewritten as:

Z (i - 1) / e^(iπ/3)

Step 2: Multiply by the Complex Conjugate

To simplify, we multiply both the numerator and the denominator by the complex conjugate of e^(iπ/3), which is e^(-iπ/3):

Z [(i - 1) * e^(-iπ/3)] / 1

This simplifies to:

Z i * e^(-iπ/3) - 1 * e^(-iπ/3)

Step 3: Expand the Numerator

Next, we expand the term e^(-iπ/3) using Euler's formula:

e^(-iπ/3) cos(-π/3) i sin(-π/3) 1/2 - i√3/2

Multiplying the terms, we get:

Z i * (1/2 - i√3/2) - (1 * (1/2 - i√3/2))

Distributing the terms, we have:

Z (i/2 - i^2√3/2) - (1/2 - i√3/2)

Since i^2 -1, this simplifies to:

Z i/2 √3/2 - 1/2 - i√3/2

Combining like terms, we get:

Z (i - i√3/2) (1/2 - √3/2)

Which can be rewritten as:

Z (1/2 - √3/2) i (1 - √3/2)

Step 4: Convert to Polar Form

To express the complex number Z in polar form, we need to find the magnitude r and the angle θ.

Step 4.1: Magnitude r

The magnitude r is given by:

r √((1/2 - √3/2)^2 (1 - √3/2)^2)

Expanding the squares:

(1/2 - √3/2)^2 1/4 - 1/2√3 3/4 1 - 1/2√3

(1 - √3/2)^2 1 - 1√3 3/4 1 - √3/2 3/4 7/4 - √3/2

Adding these:

r √(1 - 1/2√3 1 - √3/2 3/4) √2

Step 4.2: Angle θ

The angle θ is given by:

θ tan^(-1) [(1 - √3/2) / (1/2 - √3/2)]

This simplifies to:

θ tan^(-1) [2]

Final Result

Thus, the polar form of the complex number Z is:

Z √2 e^(iθ)

Where θ tan^(-1) (2).