Problem 25 - Advanced Engineering Mathematics

Question

Use \((8)\) to compute the indicated power. $$ \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)^{12} $$

Step-by-Step Solution

Answer

The answer is \(-i\).

Step 1: Identify Euler's Formula

Recall Euler's formula: \[ e^{i\theta} = \cos \theta + i \sin \theta\]Here, \(\theta\) is \(\frac{\pi}{8}\). Thus, the expression can be rewritten using Euler's formula as:\[\left(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\right) = e^{i\frac{\pi}{8}}\]

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that for any complex number \(z = r e^{i\theta}\), its \(n\)-th power is given by:\[(z)^n = r^n e^{i n \theta}\]By applying this to our expression, we have:\[\left(e^{i\frac{\pi}{8}}\right)^{12} = e^{i\frac{12\pi}{8}} = e^{i\frac{3\pi}{2}}\]

Step 3: Simplify the Result

Calculate \(e^{i\frac{3\pi}{2}}\) using Euler's formula once again:\[e^{i\frac{3\pi}{2}} = \cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}\]Now, evaluate the trigonometric functions:\(\cos \frac{3\pi}{2} = 0\) and \(\sin \frac{3\pi}{2} = -1\). Therefore:\[e^{i\frac{3\pi}{2}} = 0 + i(-1) = -i\]

Step 4: Write the Final Result

We've found that \(\left(e^{i\frac{\pi}{8}}\right)^{12} = -i\). Therefore, the original expression also equals:\[-i\]

Key Concepts

Euler's Formula

Euler's Formula is a fascinating connection between trigonometry and exponential functions involving complex numbers. It is given by the equation:\[ e^{i\theta} = \cos \theta + i \sin \theta \]Where:

• \(e\) is the base of natural logarithms, roughly equal to 2.71828.
• \(i\) is the imaginary unit, defined as \(\sqrt{-1}\).
• \(\theta\) is the angle measured in radians.

This formula shows that complex exponential functions can be expressed in terms of sines and cosines. A crucial aspect of Euler's Formula is how it simplifies the expression of complex numbers in polar form.
In our exercise, Euler's Formula allows us to represent the expression \(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\) as \(e^{i\frac{\pi}{8}}\), making it easier to handle when finding powers of the complex number, specifically by using De Moivre's Theorem.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are typically written in the form:\[ z = a + bi \]Where:

• \(a\) is the real part.
• \(b\) is the imaginary part.
• \(i\) is the imaginary unit satisfying \(i^2 = -1\).

Complex numbers can also be expressed in polar form:\[ z = r (\cos \theta + i \sin \theta) \]Here, \(r\) is the modulus (or absolute value) of the complex number, and \(\theta\) is the argument (or angle). The polar form is especially useful for multiplying and dividing complex numbers and finding their powers and roots.
In this exercise, the complex number \(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\) is a unit complex number (meaning its magnitude is 1), simplifying calculations when applying De Moivre's Theorem to find the 12th power.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In the realm of complex numbers, it plays a significant role through identities and relationships that simplify complex exponentiations.Using trigonometric identities such as:

• \(\cos\theta\) gives the real part.
• \(\sin\theta\) provides the imaginary part.

These functions can transform complex expressions into a more unified format. It's crucial to analyze how these identities allow the simplification of expressions by converting back and forth between exponential and trigonometric forms, using Euler's Formula.
In our example, after applying De Moivre's Theorem, the expression reduces to evaluating trigonometric functions: \(\cos \frac{3\pi}{2} = 0\) and \(\sin \frac{3\pi}{2} = -1\). This directly leads us to the final calculated result of \(-i\).