Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{4}$$

Step-by-Step Solution

Answer

The power in rectangular form is \( -1 \).

Step 1: Recognize Initial Complex Number

The given complex number is \[ z = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i. \]This is in the form \( a + bi \). Here, \( a = -\frac{\sqrt{2}}{2} \) and \( b = \frac{\sqrt{2}}{2} \).

Step 2: Convert to Polar Form

Calculate the magnitude of \[ |z| = \sqrt{(-\frac{\sqrt{2}}{2})^2 + (\frac{\sqrt{2}}{2})^2} = 1. \]The angle \( \theta \) (or argument) is \[ \theta = \arctan\left(\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\right) + \pi = \frac{3\pi}{4}. \]Therefore, the polar form is \[ z = 1\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right). \]

Step 3: Apply De Moivre's Theorem

We want to find \((z)^4\), so by De Moivre's Theorem,\[ z^4 = \left(1\right)^4 \left(\cos (4\cdot\frac{3\pi}{4}) + i\sin (4\cdot\frac{3\pi}{4})\right). \]Calculate\[ 4\cdot\frac{3\pi}{4} = 3\pi. \]

Step 4: Simplify the Trigonometric Expression

For \( 3\pi \),\[ \cos(3\pi) = -1 \quad \text{and} \quad \sin(3\pi) = 0. \]

Step 5: Write the Final Rectangular Form

Thus, \[ z^4 = -1 + 0i. \]So, the value of \( \left(-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\right)^4 \) in rectangular form is \[ -1. \]

Key Concepts

Rectangular Form

Complex numbers can be expressed in a form that resembles a coordinate on the complex plane, called the rectangular form. This form is expressed as \( a + bi \), where \( a \) represents the real part, and \( b \) represents the imaginary part of the complex number. For example, in the given complex number \( -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \), the real part \( a \) is \( -\frac{\sqrt{2}}{2} \) and the imaginary part \( b \) is \( \frac{\sqrt{2}}{2} \).

• The letter \( i \) is used to denote the imaginary unit, where \( i^2 = -1 \).
• Rectangular form is especially useful in addition and subtraction of complex numbers as it aligns with Cartesian coordinates.

To understand how rectangular form works with complex operations like multiplication and taking powers, we often convert these numbers into polar form.

Polar Form

Polar form is another way to express complex numbers, which highlights the magnitude and angle (often called the argument) of the complex number relative to the positive x-axis. This form is written as \( r(\cos \theta + i\sin \theta) \) or \( re^{i\theta} \), where \( r \) is the magnitude and \( \theta \) is the angle.

• The magnitude \( r \) is calculated as \( |z| = \sqrt{a^2 + b^2} \), which measures the "distance" of the complex number from the origin on the complex plane.
• The angle \( \theta \) is found using trigonometric functions usually with \( \tan^{-1}(b/a) \) and adjusting the angle based on its quadrant.

For the complex number \( -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \), the magnitude is 1, and \( \theta \) is \( \frac{3\pi}{4} \). This means the complex number can be directly expressed as \( 1(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}) \) in polar form. Polar form is particularly advantageous for multiplying complex numbers and that’s where De Moivre’s Theorem is applied.

De Moivre's Theorem

De Moivre's Theorem provides a straightforward way to raise complex numbers in polar form to any integer power. The theorem states that if \( z = r(\cos\theta + i\sin\theta) \), then \( z^n = r^n(\cos(n\theta) + i\sin(n\theta)) \).

• This theorem simplifies the process by allowing us to multiply the angle \( \theta \) by the power \( n \) and raise the magnitude \( r \) to the same power.
• It is incredibly useful when performing complex calculations that involve powers, such as \( (-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i)^4 \).

In our example, using De Moivre’s Theorem, the complex number \( 1(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}) \) raised to the fourth power becomes \( \cos(3\pi) + i\sin(3\pi) \). This evaluates to \( -1 + 0i \) in rectangular form, thus giving us the result \( -1 \).
This efficient method bypasses numerous calculations that would otherwise be required if working directly in rectangular form.