Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Find the three cube roots of \(-27 i\).

Step-by-Step Solution

Answer

The cube roots are \(\frac{3\sqrt{3}}{2} - \frac{3i}{2}\), \(3i\), and \(-\frac{3\sqrt{3}}{2} - \frac{3i}{2}\).

Step 1: Express in Polar Form

First, express the complex number \( -27i \) in polar form. Its magnitude is \( r = | -27i | = 27 \). Since it lies on the negative imaginary axis, the argument is \( \theta = -\frac{\pi}{2} \). So, the polar form is \( 27 \text{cis}(-\frac{\pi}{2}) \), where \( \text{cis} \theta = \cos \theta + i \sin \theta \).

Step 2: Use De Moivre's Theorem

To find the cube roots, use De Moivre's Theorem: if a complex number is \( r \text{cis}(\theta) \), then its cube roots are \( \sqrt[3]{r} \text{cis}\left(\frac{\theta+2k\pi}{3}\right) \) for \( k = 0, 1, 2 \). Here, \( r = 27 \) and \( \theta = -\frac{\pi}{2} \).

Step 3: Find Magnitude of Cube Root

Calculate the magnitude of the cube root: \( \sqrt[3]{27} = 3 \). This is because the cube root of 27 is simply 3.

Step 4: Find First Cube Root

For \( k = 0 \), the argument becomes \( \frac{-\frac{\pi}{2}+2(0)\pi}{3} = -\frac{\pi}{6} \). So, the first cube root is \( 3 \text{cis}(-\frac{\pi}{6}) = 3\left(\cos(-\frac{\pi}{6}) + i \sin(-\frac{\pi}{6})\right) = \frac{3\sqrt{3}}{2} - \frac{3i}{2} \).

Step 5: Find Second Cube Root

For \( k = 1 \), the argument becomes \( \frac{-\frac{\pi}{2}+2(1)\pi}{3} = \frac{\pi}{2} \). So, the second cube root is \( 3 \text{cis}(\frac{\pi}{2}) = 3\left(\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})\right) = 3i \).

Step 6: Find Third Cube Root

For \( k = 2 \), the argument becomes \( \frac{-\frac{\pi}{2}+2(2)\pi}{3} = \frac{7\pi}{6} \). So, the third cube root is \( 3 \text{cis}(\frac{7\pi}{6}) = 3\left(\cos(\frac{7\pi}{6}) + i \sin(\frac{7\pi}{6})\right) = -\frac{3\sqrt{3}}{2} - \frac{3i}{2} \).

Key Concepts

Polar Form

Polar form is a way to represent complex numbers using a combination of a magnitude and an angle. Complex numbers have both a real part and an imaginary part. However, in polar form, they are expressed as \[ r \text{cis}(\theta) = r (\cos(\theta) + i \sin(\theta)) \]. This form is particularly useful in simplifying the process of finding powers and roots of complex numbers.

• Magnitude \( r \): This is the absolute value of the complex number, which is the distance from the origin in a complex plane.
• Angle \( \theta \): This is known as the argument of the complex number, which represents the angle formed between the positive real axis and the line connecting the origin to the number.

When we find the cube roots of a complex number like \(-27i\), expressing it in polar form simplifies calculations and helps us visualize roots in the complex plane.

Complex Numbers

Complex numbers are numbers composed of a real part and an imaginary part. They are of the form \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary component multiplied by \( i \), the imaginary unit.

• Imaginary unit \( i \): Defined as \( i^2 = -1 \).
• Real part: The component \( a \), representing the horizontal axis in a complex plane.
• Imaginary part: The component \( b \), representing the vertical axis in a complex plane.

Converting the complex number \(-27i\) into polar form helps with operations like finding powers and roots through straightforward trigonometric manipulation, avoiding cumbersome algebraic methods.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for working with complex numbers in polar form. It provides a direct method for finding powers and roots of complex numbers.
The theorem states that for a complex number in polar form, \( r \text{cis}(\theta) \), its \( n^{th} \) power is given by \((r \text{cis}(\theta))^n = r^n \text{cis}(n \theta)\).
When applied to roots, namely the cube root in this exercise, De Moivre's states:\[\sqrt[3]{r} \text{cis}\left(\frac{\theta + 2k\pi}{3}\right),\quad \text{for } k = 0, 1, 2.\]This provides us clear steps to find all cube roots, by incrementing \( k \) and properly adjusting the argument.

Magnitude

Magnitude, or absolute value, of a complex number \( z = a + bi \) is a measure of its distance from the origin in the complex plane.
It can be found using the formula:\[ |z| = \sqrt{a^2 + b^2} \]This distance is crucial in polar form.

• For \(-27i\), the magnitude is \( 27 \), since only the imaginary part contributes: \( \sqrt{0^2 + (-27)^2} = 27 \).
• Magnitude defines the size of circles around the origin through which roots like cube roots are located.

Understanding the magnitude helps determine how far from the origin the cube roots will be positioned.

Trigonometric Form

Trigonometric form is a framework used to express complex numbers, similar to polar form but specifically focusing on sine and cosine components.
This form is central when expressing complex numbers as \( a + bi = r(\cos(\theta) + i\sin(\theta)) \).The trigonometric form is particularly useful in converting back to the standard \( a + bi \) notation after calculations.

• Conversion: Makes it easy to multiply or divide complex numbers.
• Cube Roots: Involves finding the trigonometric expression, followed by using angles \(-\frac{\pi}{6}\), \(\frac{\pi}{2}\), and \(\frac{7\pi}{6}\) to find the cubes roots in the example.

This approach helps visualize operations performed on complex numbers in trigonometric form, leveraging familiar sine and cosine functions.