Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i\right)^{8}$$

Step-by-Step Solution

Answer

The power is 1, in rectangular form.

Step 1: Identify the Complex Number

The given complex number is \( z = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i \). This is written in rectangular form as \( z = a + bi \), where \( a = \frac{\sqrt{2}}{2} \) and \( b = -\frac{\sqrt{2}}{2} \).

Step 2: Convert to Polar Form

To convert to polar form, determine the magnitude \( r \) and the angle \( \theta \). The magnitude is calculated as:\[ r = \sqrt{a^2 + b^2} = \sqrt{\left(\frac{\sqrt{2}}{2}\right)^2 + \left(-\frac{\sqrt{2}}{2}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = 1 \]The angle \( \theta \) is calculated using the arctangent of \( \frac{b}{a} \):\[ \theta = \tan^{-1}\left(\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \]Thus, the polar form is \( z = 1(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})) \).

Step 3: Use De Moivre's Theorem

De Moivre's Theorem states that: \[ z^n = r^n \left( \cos(n\theta) + i\sin(n\theta) \right) \]For \( n = 8 \), we have:\[ z^8 = 1^8 \left( \cos(8 \times -\frac{\pi}{4}) + i\sin(8 \times -\frac{\pi}{4}) \right) \]This simplifies to:\[ z^8 = \cos(-2\pi) + i\sin(-2\pi) \]

Step 4: Simplify the Trigonometric Expressions

We know that \( \cos(-2\pi) = \cos(0) = 1 \) and \( \sin(-2\pi) = \sin(0) = 0 \). Therefore,\[ z^8 = 1 + 0i \]

Step 5: Write the Answer in Rectangular Form

The expression \( z^8 = 1 + 0i \) is already in rectangular form. This means our solution is simply \( 1 \).

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical entities that extend the idea of the one-dimensional number line to a two-dimensional complex plane. They consist of two parts: a real part and an imaginary part. Complex numbers are expressed in the form \( z = a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part. The unit imaginary number is denoted as \( i \), which satisfies the equation \( i^2 = -1 \).

• Understanding complex numbers helps in solving equations where the solution is not possible with just real numbers.
• They are widely used in engineering, physics, and applied sciences due to their ability to simplify the representation of periodic functions.

Consider \( z = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i \). Here, \( \frac{\sqrt{2}}{2} \) is the real part, and \( -\frac{\sqrt{2}}{2} \) is the imaginary part, reflecting its position in the complex plane.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number theory, particularly useful for finding powers and roots of complex numbers. The theorem states: \[(z^n = r^n \left( \cos(n\theta) + i\sin(n\theta) \right) )\] where \( r \) is the magnitude of the complex number and \( \theta \) is the angle in polar form. De Moivre's Theorem simplifies the process of raising a complex number to a power.

• This theorem is particularly useful since it allows computation with trigonometric forms rather than complex numbers in standard rectangular form.
• Using this theorem, the challenge of calculating powers and roots becomes a simple matter of multiplication and division of angles, as well as exponentiation of magnitude.

In our exercise, raising \( z = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i \) to the power of 8 involves first converting to polar form, utilizing \( r = 1, \theta = -\frac{\pi}{4} \), and then applying De Moivre's Theorem to find \( z^8 = \cos(-2\pi) + i\sin(-2\pi) \).

Polar Form

The polar form of a complex number provides a different way of understanding and representing complex numbers. In polar form, a complex number is expressed as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) of the complex number and \( \theta \) is the argument (or angle).

• The magnitude \( r \) is calculated with the formula \( r = \sqrt{a^2 + b^2} \).
• The angle \( \theta \) is determined by \( \tan^{-1}(b/a) \).
• The polar form elegantly ties trigonometry with complex numbers, facilitating operations such as multiplication, division, and power of complex numbers.

In our exercise, for the complex number \( z = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i \), the magnitude is \( r = 1 \) and the angle \( \theta = -\frac{\pi}{4} \). Thus, the polar form becomes \( z = 1(\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4})) \).

Trigonometric Expressions

Trigonometric expressions play a vital role in converting complex numbers from polar form to rectangular form and vice versa. These expressions are critical when using De Moivre's Theorem to manipulate complex numbers.

• The key trigonometric functions used are sine \( \sin \theta \) and cosine \( \cos \theta \) which help in locating the complex number on the unit circle.
• The conversion between polar and rectangular forms relies on these trigonometric identities.

In our solution, simplifying the trigonometric expressions \( \cos(-2\pi) \) and \( \sin(-2\pi) \) utilizes the periodic nature of trigonometric functions:

• \( \cos(-2\pi) = \cos(0) = 1 \)
• \( \sin(-2\pi) = \sin(0) = 0 \)

This simplification is essential in determining that \( z^8 = 1 + 0i \), bringing the expression back to the rectangular form.