Problem 79 - Algebra and Trigonometry

Question

Find the indicated power using De Moivre's Theorem. $$ (2 \sqrt{3}+2 i)^{-5} $$

Step-by-Step Solution

Answer

\(-\sqrt{3}/2048 - i/2048\)

Step 1: Convert to Polar Form

First, we need to express the complex number \(2\sqrt{3} + 2i\) in polar form. Start by finding the modulus (r) using the formula \( r = \sqrt{a^2 + b^2} \) where \( a = 2\sqrt{3} \) and \( b = 2 \). Thus, \( r = \sqrt{(2\sqrt{3})^2 + 2^2} = \sqrt{12 + 4} = \sqrt{16} = 4 \).

Step 2: Determine the Argument

The argument (\(\theta\)) of the complex number is determined using \( \tan^{-1}(b/a) \), giving: \( \theta = \tan^{-1}(2/(2\sqrt{3})) = \tan^{-1}(1/\sqrt{3}) = \pi/6 \). Therefore, the polar form of the complex number is \( 4 \text{cis} \pi/6 \).

Step 3: Apply De Moivre's Theorem

De Moivre's Theorem states that for a complex number \( r \text{cis} \theta \), its nth power is \( r^n \text{cis}(n\theta) \). For \(n = -5\), we have: \( (4 \text{cis} \pi/6)^{-5} = 4^{-5} \text{cis}(-5\cdot \pi/6) \).

Step 4: Calculate the Power of the Modulus

Compute \( 4^{-5} \), which is \( 1/4^5 = 1/1024 \).

Step 5: Calculate the Power of the Argument

Compute \( -5 \cdot \pi/6 = -5\pi/6 \). Thus, the resulting complex number in polar form is \( (1/1024) \text{cis}(-5\pi/6) \).

Step 6: Convert Back to Rectangular Form

Use \( \text{cis}(-5\pi/6) = \cos(-5\pi/6) + i\sin(-5\pi/6) \) to convert back to rectangular form. From trigonometry, \( \cos(-5\pi/6) = -\sqrt{3}/2 \) and \( \sin(-5\pi/6) = -1/2 \). Thus, the rectangular form is \( (1/1024)(-\sqrt{3}/2 - i/2) = -\sqrt{3}/2048 - i/2048 \).

Key Concepts

Complex Numbers

Complex numbers are fascinating because they expand our understanding of numbers beyond the real line into a plane of existence known as the complex plane. They are of the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) representing the imaginary unit satisfying \( i^2 = -1 \).

• These numbers can be visualized as points or vectors on the complex plane.
• The horizontal axis is known as the real axis, while the vertical axis is called the imaginary axis.

In this configuration, each complex number corresponds to a unique point or vector in the plane. This enables us to perform operations like addition, subtraction, multiplication, and division in a manner that is analogous to operations on real numbers, but adjusted for the complex components.

Polar Form

The polar form of a complex number is another way to express these numbers, using their distance from the origin (modulus) and the angle with the positive real axis (argument). This form is incredibly useful when dealing with complex number powers and roots, simplifying many calculations.

• Polar form is represented as \( r \text{cis} \theta \), where \( r \) is the modulus, and \( \theta \) is the argument.
• The expression \( \text{cis} \theta \) stands for \( \cos \theta + i \sin \theta \).

Converting a complex number into polar form requires calculating both the modulus and argument:

• The modulus \( r \) is the distance of the point from the origin, given by \( r = \sqrt{a^2 + b^2} \).
• The argument \( \theta \) is the angle formed with the real axis, found using \( \theta = \tan^{-1}(b/a) \).

Modulus and Argument

The modulus and argument are fundamental in shifting between the rectangular and polar forms of complex numbers. This conversion can be particularly significant when applying De Moivre's Theorem, which relies on polar form.
The **modulus** \( r \) of a complex number \( a + bi \) is a measure of its "size" or