Problem 72 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The answer is 16.

Step 1: Convert to Polar Form

The complex number \(1 - i\) needs to be expressed in polar form \(r(\cos \theta + i \sin \theta)\). First calculate the modulus: \(r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\). Then find the argument \(\theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}\). So, \(1 - i = \sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right)\right)\).

Step 2: Apply De Moivre’s Theorem

De Moivre’s Theorem states that \((r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i \sin(n\theta))\). Using this theorem, compute \((1-i)^{8} = (\sqrt{2})^8 \left(\cos\left(8 \times -\frac{\pi}{4}\right) + i \sin\left(8 \times -\frac{\pi}{4}\right)\right)\).

Step 3: Simplify the Modulus

Calculate \((\sqrt{2})^8\). This equals \( (2^{1/2})^8 = 2^{4} = 16\). So, the modulus is 16.

Step 4: Simplify the Argument

Calculate \( 8 \times -\frac{\pi}{4} = -2\pi \). Since cosine and sine of any integer multiple of \(2\pi\) are \(1\) and \(0\) respectively, we plug these into the expression.

Step 5: Write Final Result in Rectangular Form

From the calculations, we have \(16(\cos(-2\pi) + i \sin(-2\pi)) = 16(1 + 0i) = 16\).

Key Concepts

Complex Numbers

Complex numbers are numbers that extend the idea of traditional numbers by introducing an imaginary unit, denoted as \(i\), where \(i^2 = -1\). This allows the combination of real and imaginary components in a number. A complex number takes the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
These numbers are particularly useful in fields such as engineering and physics, as they allow for a more comprehensive representation of signals and other elements. When working with complex numbers, it is important to understand their basic operations,

• Addition: Combine the real parts and the imaginary parts separately.
• Multiplication: Use the distributive property, remembering that \(i^2 = -1\).
• Conjugate: The conjugate of \(a + bi\) is \(a - bi\), often used to rationalize denominators.

Grasping these basics will set the foundation for more advanced calculations involving complex numbers.

Polar Form

The polar form of complex numbers is a way to express them in terms of magnitude and angle, rather than real and imaginary components. This form is very useful when dealing with powers and roots of complex numbers. A complex number \(a + bi\) is converted to polar form \(r(\cos \theta + i \sin \theta)\), where

• \(r\) is the modulus, representing the distance from the origin to the point \((a, b)\) in the complex plane.
• \(\theta\) is the argument, representing the angle formed with the positive x-axis.

To convert a complex number into its polar form, first find

• the modulus \(r = \sqrt{a^2 + b^2}\),
• the argument \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

Understanding the polar form is essential when applying De Moivre's Theorem, which simplifies raising complex numbers to powers.

Trigonometry

Trigonometry in the context of complex numbers involves using trigonometric functions like cosine and sine to describe the position of a complex number in the polar coordinate system. These functions help describe the angle and orientation of a complex number:

• Cosine (\(\cos\theta\)): Represents the x-coordinate (adjacent side) in the unit circle.
• Sine (\(\sin\theta\)): Represents the y-coordinate (opposite side) in the unit circle.

In the polar form \(r(\cos \theta + i \sin \theta)\), \(\cos \theta\) and \(\sin \theta\) give the direction of the vector representing the complex number in the complex plane.
These trigonometric expressions simplify operations such as multiplication, division, and finding powers of complex numbers when combined with De Moivre's Theorem, which leverages the properties of trigonometric functions.

Modulus and Argument

The modulus and argument are key concepts when expressing complex numbers in polar form. They provide a way to understand the position and magnitude of a complex number
in the complex plane without directly referencing its real and imaginary components.

• Modulus (\(r\)): This is the distance from the origin to the point representing the complex number and is calculated as \(r = \sqrt{a^2 + b^2}\). It provides the absolute size of the complex number irrespective of direction.
• Argument (\(\theta\)): This is the angle formed with the positive x-axis and measured counterclockwise, found using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). The argument specifies the direction of the complex number.

Together, the modulus and argument allow us to rewrite complex numbers in polar form, assisting with more complex operations such as exponentiation through De Moivre's Theorem.