Problem 27 - Advanced Engineering Mathematics

Question

$$8^{1 / 3}=2\left[\cos \frac{2 k \pi}{3}+i \sin \frac{2 k \pi}{3}\right], \quad k=0,1,2$$ $$w_{0}=2[\cos 0+i \sin 0]=2 ; \quad w_{1}=2\left[\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right]=-1+\sqrt{3} i$$ $$w_{2}=2\left[\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right]=-1-\sqrt{3} i$$

Step-by-Step Solution

Answer

The cube roots of 8 are: 2, \(-1 + \sqrt{3}i\), \(-1 - \sqrt{3}i\).

Step 1: Understand the Expression

The given expression represents the cube roots of the number 8 in complex form, using De Moivre's Theorem. The number 8 is expressed in the form of its cube roots as: \[8^{1/3} = 2 \left[\cos \frac{2k\pi}{3} + i \sin \frac{2k\pi}{3}\right],\] where \(k=0, 1, 2\). This formula indicates that there are three distinct cube roots, obtained by substituting the values of \(k\) into the trigonometric functions.

Step 2: Calculate the Roots for k=0

Substitute \(k=0\) into the formula:\[w_0 = 2[\cos 0 + i \sin 0] = 2[1 + i \, 0] = 2.\]The calculation shows that the first root is \(w_0 = 2\).

Step 3: Calculate the Roots for k=1

Substitute \(k=1\) into the formula:\[w_1 = 2 \left[\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right].\]Using the trigonometric identities: \(\cos \frac{2\pi}{3} = -\frac{1}{2}\) and \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\), \[w_1 = 2\left[-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right] = -1 + \sqrt{3} i.\]

Step 4: Calculate the Roots for k=2

Substitute \(k=2\) into the formula:\[w_2 = 2 \left[\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}\right].\]Using the trigonometric identities: \(\cos \frac{4\pi}{3} = -\frac{1}{2}\) and \(\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}\), \[w_2 = 2\left[-\frac{1}{2} - i \frac{\sqrt{3}}{2}\right] = -1 - \sqrt{3} i.\]

Step 5: Summarize the Roots

The cube roots of 8 in complex form are \[w_0 = 2, \ w_1 = -1 + \sqrt{3}i, \ w_2 = -1 - \sqrt{3}i.\]Each represents one of the possible solutions of the equation \(z^3 = 8\) in the complex plane.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful method for finding powers and roots of complex numbers.
It states that for any complex number written in polar form, \( z = r(\cos \theta + i \sin \theta) \), its \( n \)-th power can be calculated using:

• \( z^n = r^n \left( \cos(n \theta) + i \sin(n \theta) \right). \)

This theorem simplifies computations involving powers and roots, transforming them into straightforward trigonometric calculations.
It also bridges complex numbers and trigonometry, showing that complex numbers can be represented as rotated vectors in the complex plane. In our exercise, we used it to find cube roots by rewriting 8 as a complex number in polar form and applying the theorem with \( n = \frac{1}{3} \). The three roots provided reflect the symmetry and periodicity of the trigonometric functions involved.

Cube Roots of Unity

The cube roots of unity are solutions to the equation \( z^3 = 1 \).
These roots are typically expressed using complex numbers and are important because they exhibit symmetry on the unit circle in the complex plane.
Importantly, they demonstrate rotational symmetry, equally dividing the circle.

• The three roots are: \( 1 \), \( \omega \), and \( \omega^2 \),
• where \( \omega = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \) and \( \omega^2 = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \).

These are fascinating because when you multiply any of them together or raise them to a power, you still get one of the roots as the result.
This property makes them crucial in simplifying polynomial equations and finding roots of unity. In the exercise, we calculated the complex cube roots of 8, which follow similar properties and reveal the elegance of the complex plane.

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, allowing us to transform trigonometric expressions into equivalent forms.
They are especially handy when working with complex numbers in polar form, as they help simplify calculations.
Understanding key identities aids in applying De Moivre's Theorem efficiently.

• The most commonly used identities include the Pythagorean Identities, like \( \cos^2\theta + \sin^2\theta = 1 \), and angle sum formulas such as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
• For the specific angles in our problem, the values \( \cos \frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \) are derived using the symmetrical properties of the unit circle.

In this exercise, these identities help find precise trigonometric values that lead to the correct computation of the complex cube roots of any number such as 8. This showcases their vital role in navigating through problems involving complex numbers.