Problem 69 - Algebra and Trigonometry

Question

\(65-76=\) Find the indicated power using DeMoivre's Theorem. $$ \left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{12} $$

Step-by-Step Solution

Answer

The result is \(-1\).

Step 1: Identify the Complex Number

The complex number given is \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i \). We need to convert it to polar form for easier computation.

Step 2: Convert to Polar Form

The modulus of the complex number is \( r = \sqrt{ \left( \frac{\sqrt{2}}{2} \right)^2 + \left( \frac{\sqrt{2}}{2} \right)^2 } = 1 \). The argument \( \theta \) such that \( \cos(\theta) = \frac{\sqrt{2}}{2} \) and \( \sin(\theta) = \frac{\sqrt{2}}{2} \) is \( \theta = \frac{\pi}{4} \). Thus, the polar form is \( 1 \text{cis} \frac{\pi}{4} \).

Step 3: Apply DeMoivre's Theorem

DeMoivre's Theorem states that \( (r \text{cis} \theta)^n = r^n \text{cis} (n\theta) \). Here, \( n = 12 \), \( r = 1 \), and \( \theta = \frac{\pi}{4} \). Therefore, the expression becomes \( 1^{12} \text{cis} (12 \cdot \frac{\pi}{4}) = \text{cis} 3\pi \).

Step 4: Simplify the Argument

The angle \( 3\pi \) is equivalent to \( 3\pi - 2\pi = \pi \) since we can subtract \( 2\pi \) to find a coterminal angle. Therefore, the expression simplifies to \( \text{cis} \pi \), which in rectangular form is \( \cos(\pi) + i \sin(\pi) = -1 + 0i \).

Key Concepts

Complex Numbers

Complex numbers extend the idea of one-dimensional numbers, like real numbers, to two dimensions by including imaginary components. They are typically written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part of the complex number.

The imaginary unit \( i \) is defined as \( i^2 = -1 \). This allows for calculations that are not possible with just real numbers.

• Real Part: The coefficient \( a \) of a complex number \( a + bi \).
• Imaginary Part: The coefficient \( b \) in the complex number \( a + bi \).

Understanding these basic components allows us to perform various operations on complex numbers, including addition, subtraction, and manipulation into other forms like polar form.

Polar Form

Complex numbers can also be represented in polar form, which can sometimes simplify the process of multiplication and division. The polar form emphasizes the modulus and the argument of a complex number.

A complex number in polar form is represented as \( r \text{cis} \theta \), where \( r \) is the modulus and \( \theta \) is the argument.

• The term \( \text{cis} \theta \) is a shorthand notation for \( \cos \theta + i \sin \theta \).
• Polar form is particularly useful for applying DeMoivre's Theorem, which simplifies the calculation of powers and roots of complex numbers.

This form of representing complex numbers leverages the geometric interpretation of the complex plane, where the movement from the origin is described in terms of direction and distance.

Modulus and Argument

In polar form, two important concepts are the modulus and the argument of a complex number. The modulus, sometimes referred to as the magnitude, represents the distance from the origin to the point in the complex plane.

• The modulus \( r \) of a complex number \( a + bi \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \).
• The argument \( \theta \) is the angle formed with the positive real axis and is measured counterclockwise. It is often determined using trigonometric ratios: \( \theta = \tan^{-1}(\frac{b}{a}) \).

Understanding these two elements is crucial for converting complex numbers into polar form and for comprehending the operation of multiplying and exponentiating complex numbers through DeMoivre's Theorem.

Rectangular Form

While polar form is very powerful, complex numbers are most commonly written in rectangular form as \( a + bi \). This form emphasizes the coordinates of the complex number on the complex plane with a real part and an imaginary part.

Rectangular form is particularly useful for straightforward addition and subtraction of complex numbers because like terms (real with real and imaginary with imaginary) are easily combined.

• Addition: \( (a + bi) + (c + di) = (a + c) + (b + d)i \)
• Subtraction: \( (a + bi) - (c + di) = (a - c) + (b - d)i \)

Converting between rectangular and polar forms allows for different mathematical operations to be more easily executed, depending on what the problem requires. Both forms provide a complete picture of any given complex number.