Problem 24 - Advanced Engineering Mathematics

Question

$$(2 \sqrt{2})^{4}\left[\cos \frac{8 \pi}{3}+i \sin \frac{8 \pi}{3}\right]=-32+32 \sqrt{3} i$$

Step-by-Step Solution

Answer

The expression simplifies to \(-32 + 32\sqrt{3}i\), confirming the given result.

Step 1: Understanding the Expression

The expression \((2 \sqrt{2})^{4}\left[\cos \frac{8\pi}{3} + i \sin \frac{8\pi}{3}\right]\) represents a complex number in polar form, where \(r = (2 \sqrt{2})^{4}\) and \(\theta = \frac{8\pi}{3}\). We need to convert this polar form to a rectangular (Cartesian) form.

Step 2: Compute the Modulus

Compute the modulus \(r = (2 \sqrt{2})^{4}\).1. Calculate \(2 \sqrt{2}\) squared: \((2 \sqrt{2})^2 = 8\).2. Raise the result to the power of 2: \(8^2 = 64\).Therefore, \(r = 64\).

Step 3: Simplify the Argument

Determine the equivalent angle for \(\frac{8\pi}{3}\):1. Find a coterminal angle by subtracting \(2\pi\): \(\frac{8\pi}{3} - 2\pi = \frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3}\).

Step 4: Convert to Rectangular Form

Convert the complex number from polar to rectangular form:1. Calculate \(\cos \frac{2\pi}{3}\): \(\cos \frac{2\pi}{3} = -\frac{1}{2}\).2. Calculate \(\sin \frac{2\pi}{3}\): \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\).3. Combine with modulus: \[ 64 \left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right) = 64 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) \] \[ = 64 \times -\frac{1}{2} + 64 \times i \frac{\sqrt{3}}{2} \] \[ = -32 + 32\sqrt{3}i \].

Step 5: Verify Against Given Answer

Compare the calculated result \(-32 + 32\sqrt{3}i\) with the given expression. They match, confirming our solution is correct.

Key Concepts

Polar Form

Let's start with understanding the polar form of complex numbers. Polar form is a way of expressing complex numbers where each number is represented by a modulus (or magnitude) and an argument (angle). It is often used when dealing with multiplication and division of complex numbers, due to its simplified representation.

In polar form, a complex number is expressed as:

• \[ z = r (\cos \theta + i \sin \theta) \]

where:

• \( r \) is the modulus
• \( \theta \) is the argument

This form is compact and perfect for transformations between different expressions of complex numbers. The given exercise uses polar form to represent \( (2 \sqrt{2})^4[\cos \frac{8\pi}{3} + i \sin \frac{8\pi}{3}] \).

Rectangular Form

Rectangular form, or Cartesian form, is a straightforward way to represent complex numbers. Here, a complex number is written in terms of its real and imaginary parts:

• \[ z = a + bi \]

where:

• \( a \) is the real part
• \( b \) is the imaginary part

In the exercise, the task is to convert the given polar form into rectangular form. This involves calculating both the cosine and sine of the argument and multiplying by the modulus. Through conversion, the polar representation \( (2 \sqrt{2})^4[\cos \frac{8\pi}{3} + i \sin \frac{8\pi}{3}] \) simplifies and leads us to the rectangular form \( -32 + 32\sqrt{3}i \).

Modulus and Argument

Modulus and argument are vital components in understanding complex numbers in polar form. The modulus of a complex number is simply its distance from the origin in the complex plane and it is always a positive number.

• The modulus \( r \) can be derived from the real and imaginary parts using: \[ r = \sqrt{a^2 + b^2} \]

The argument is the angle that the line representing the complex number makes with the positive real axis. This exercise specifically requires simplification of the argument \( \theta = \frac{8\pi}{3} \). By reducing this angle to \( \frac{2\pi}{3} \) through coterminal angle calculations, all operations become basic trigonometric functions.

Trigonometric Representation

Trigonometric representation ties everything together, providing a deeper understanding of how to move from polar to rectangular form and vice versa. This representation makes use of the unit circle and standard trigonometric identities.

• \( \cos \theta \) and \( \sin \theta \) are the projections on the x-axis and y-axis, respectively.

In the solution provided, \( \cos \frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \). These values are derived from standard trigonometric values on the unit circle.

Trigonometric representation is particularly beneficial for understanding complex numbers visually, and it aids in recognizing patterns through the symmetry and properties of sin and cos functions. Truly grasping this concept expands the sphere of possible transformations between different forms, enhancing fluency in complex number arithmetic.