Problem 1 - Advanced Engineering Mathematics

Question

$$e^{\frac{\pi}{6} i}=\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}=\frac{\sqrt{3}}{2}+\frac{1}{2} i$$

Step-by-Step Solution

Answer

Yes, the expression is verified correctly using Euler's formula.

Step 1: Understand the Problem

We want to verify the equality given: \( e^{\frac{\pi}{6} i} = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \), and show it equals \( \frac{\sqrt{3}}{2} + \frac{1}{2} i \). This involves using Euler's formula: \( e^{ix} = \cos x + i \sin x \).

Step 2: Apply Euler's Formula

Recall Euler's formula: \( e^{ix} = \cos x + i \sin x \). For \( x = \frac{\pi}{6} \), this becomes \( e^{\frac{\pi}{6} i} = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \). Substitute \( x = \frac{\pi}{6} \) into this formula.

Step 3: Calculate \( \cos \frac{\pi}{6} \) and \( \sin \frac{\pi}{6} \)

From trigonometric values: \( \cos \frac{\pi}{6} \) is known to be \( \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} \) is \( \frac{1}{2} \). Therefore, \( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} = \frac{\sqrt{3}}{2} + \frac{1}{2} i \).

Step 4: Verify the Equality

The expression \( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} = \frac{\sqrt{3}}{2} + \frac{1}{2} i \) matches exactly with the given expression in the problem. This confirms that \( e^{\frac{\pi}{6} i} = \frac{\sqrt{3}}{2} + \frac{1}{2} i \) as expected.

Key Concepts

Complex Numbers

Complex numbers are a major cornerstone of advanced mathematics, expanding our understanding of numbers beyond the real number line. Essentially, a complex number is a number that has two components: a real part and an imaginary part. It is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The imaginary unit \( i \) has a unique property where \( i^2 = -1 \). Here's why complex numbers are important:

• They allow us to solve equations that have no real solutions, like \( x^2 + 1 = 0 \).
• They provide a complete field for algebraic operations.
• They're crucial in various fields such as engineering, physics, and computer science.

A beautiful way complex numbers interact with mathematics is through Euler's formula, which connects them with trigonometric and exponential functions. Understanding Euler's formula deeply involves recognizing the significance of complex numbers and how they beautifully bridge different areas of math.

Trigonometric Functions

Trigonometric functions, primarily sine and cosine, are fundamental concepts not only in geometry but also in many areas of mathematics, including calculus and complex analysis. They are particularly crucial when we deal with periodic phenomena, such as waves and oscillations. In trigonometry, the sine and cosine of an angle correspond to the y-coordinate and x-coordinate of a point on the unit circle. For the angle \( \frac{\pi}{6} \), these trigonometric values are:

• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sin \frac{\pi}{6} = \frac{1}{2} \)

These values are not arbitrary—they can be derived from the unit circle and the geometry of equilateral triangles. In the context of Euler's formula, these functions allow us to express complex exponentials in terms of real-valued functions, providing insight into the intrinsic nature of oscillations and waves through complex numbers.

Exponential Functions

Exponential functions are pivotal in mathematics, often serving as models for growth, decay, and complex oscillations. The most notable exponential function is the natural exponential function \( e^x \), where \( e \) is approximately 2.718, known as Euler's number. Exponential functions have interesting properties, such as the ability for any exponential function of the form \( e^{ix} \) to be expressed in terms of trigonometric functions via Euler’s formula:\[ e^{ix} = \cos x + i\sin x \]This mathematics reveals a fascinating link between exponential and trigonometric functions. The formula shows that while exponential functions grow or decay in real numbers, when extended to imaginary numbers, they describe rotations and oscillations.This characteristic is vital in fields like signal processing and quantum mechanics, where oscillatory behaviors are pivotal. Understanding how exponential functions can transform into trigonometric expressions through Euler's formula is essential for grasping the deeper ties between different mathematical disciplines.