Problem 72 - Algebra and Trigonometry

Question

Find the indicated power using De Moivre's Theorem. $$ (1-i)^{8} $$

Step-by-Step Solution

Answer

The result of \((1-i)^8\) is 16.

Step 1: Convert to Polar Form

First, convert the complex number \(1-i\) into its polar form. The modulus \(r\) is computed as \(\sqrt{1^2 + (-1)^2} = \sqrt{2}\). The argument \(\theta\) can be found using \(\tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}\). Thus, \(1-i = \sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\).

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, to raise a complex number to a power, use \([r(\cos \theta + i\sin \theta)]^n = r^n[\cos(n\theta) + i\sin(n\theta)]\). For \((1-i)^8\), \(r = \sqrt{2}\) and \(\theta = -\frac{\pi}{4}\), so we have: \(\sqrt{2}^8 \left(\cos\left(8 \times -\frac{\pi}{4}\right) + i\sin\left(8 \times -\frac{\pi}{4}\right)\right)\).

Step 3: Simplify the Exponent and Angle

Calculate \(\sqrt{2}^8 = (2)^4 = 16\). The angle becomes \(8 \times -\frac{\pi}{4} = -2\pi\). So, we need to evaluate \(\cos(-2\pi) + i\sin(-2\pi)\).

Step 4: Evaluate Trigonometric Functions

The values of the trigonometric functions for \(-2\pi\) are \(\cos(-2\pi) = 1\) and \(\sin(-2\pi) = 0\). Substituting these back, we have: \(16(1 + 0i) = 16\).

Step 5: Write the Final Result

Thus, \((1-i)^8 = 16\), which is the final real result with no imaginary part.

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and have the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, defined as \(i^2 = -1\).
These numbers are often used in advanced mathematics because they allow us to solve equations that do not have real solutions, such as those involving the square root of a negative number.
The real part of a complex number is \(a\), and the imaginary part is \(b\).

• For example, in the complex number \(1 - i\), 1 is the real part and \(-1\) is the imaginary part.
• On the complex plane, complex numbers are represented as points with coordinates \((a, b)\).

Understanding complex numbers is crucial when dealing with functions that describe periodic behavior, such as in engineering and physics.

Polar Form

The polar form of a complex number is a way to express it using a modulus \(r\) (distance from the origin) and an argument \(\theta\) (angle with the positive x-axis).
This form is particularly useful when multiplying, dividing, or raising complex numbers to a power.
In polar form, a complex number \(a + bi\) converts to \(r(\cos \theta + i\sin \theta)\).

• The modulus is calculated as \(r = \sqrt{a^2 + b^2}\).
• The argument is determined by \(\theta = \tan^{-1}(\frac{b}{a})\).

By expressing complex numbers in their polar form, operations such as finding powers or roots become straightforward through De Moivre's Theorem.

Trigonometric Functions

Trigonometric functions involve angles and the relationships between the sides of triangles.
They are based on the unit circle, a circle with a radius of 1 centered at the origin of the coordinate plane.
The cosine and sine functions correspond to the x and y coordinates of a point on the unit circle.

• For the angle \(-\frac{\pi}{4}\), \(\cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\) and \(\sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\).
• For the angle \(-2\pi\), \(\cos(-2\pi) = 1\) and \(\sin(-2\pi) = 0\).

These functions are periodic, repeating their values over regular intervals. Understanding how to evaluate these functions is key to applying De Moivre's Theorem and working with complex numbers in polar form.

Mathematical Theorem

A mathematical theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted operations.De Moivre's Theorem is a powerful tool in the field of complex numbers, named after the French mathematician Abraham de Moivre. It states that for a complex number \((r \cos \theta + i \sin \theta)\), raising it to a power \(n\) is given by:\[(r^n)(\cos(n\theta) + i\sin(n\theta))\]

• This theorem simplifies the process of finding powers of complex numbers when they are in polar form.
• It illustrates the relationship between exponentiation and rotation in the complex plane.

Applications of De Moivre's Theorem extend to solving polynomial equations and performing various engineering and physics calculations that involve periodic phenomena.