Problem 12 - Advanced Engineering Mathematics

Question

$$e^{5+\frac{5 \pi}{2} i}=e^{5} \cos \frac{5 \pi}{2}+i e^{5} \sin \frac{5 \pi}{2}=e^{5} i$$

Step-by-Step Solution

Answer

The expression simplifies to \( e^5 i \).

Step 1: Understand the Problem

The given expression involves Euler's formula for complex exponentials. We need to compute \( e^{5 + \frac{5\pi}{2} i} \) and verify the equality.

Step 2: Apply Euler's Formula

Euler's formula states that \( e^{ix} = \cos(x) + i\sin(x) \). For an expression of the form \( e^{a + bi} \), it equals \( e^a(\cos(b) + i\sin(b)) \). Here, \( a = 5 \) and \( b = \frac{5\pi}{2} \).

Step 3: Calculate the Real Part

Using \( \cos(x) = \cos(x + 2\pi k) \), we simplify \( \cos\left(\frac{5\pi}{2}\right) \). It's equivalent to \( \cos\left(\frac{\pi}{2}\right) \). Hence, \( \cos\left(\frac{5\pi}{2}\right) = 0 \).

Step 4: Calculate the Imaginary Part

Similarly, \( \sin(x) = \sin(x + 2\pi k) \), thus \( \sin\left(\frac{5\pi}{2}\right) \) simplifies to \( \sin\left(\frac{\pi}{2}\right) \). Thus, \( \sin\left(\frac{5\pi}{2}\right) = 1 \).

Step 5: Write the Complex Number

Plug the values back into the Euler's formula expression: \( e^{5} \left( \cos\left(\frac{5\pi}{2}\right) + i\sin\left(\frac{5\pi}{2}\right) \right) = e^{5}(0 + i\cdot 1) = e^5 i \).

Key Concepts

Euler's Formula

Euler's Formula is an important and fascinating equation in mathematics. It connects the world of complex numbers and trigonometric functions. The formula is expressed as \( e^{ix} = \cos(x) + i\sin(x) \). This is particularly useful in the analysis of complex exponential functions.

• The formula shows how complex exponentials relate to trigonometric functions.
• It forms the basis for tackling problems involving complex numbers and their exponential forms.

This relationship is deeply rooted in the oscillatory nature of trigonometric functions, and it allows mathematicians and engineers to work across different domains smoothly. When a complex number is in the form \( e^{a+bi} \), Euler's formula helps break it down into a simpler form using \( e^a(\cos(b) + i\sin(b)) \). This brings the intuitive geometry of circles (via sin and cos) into the algebraic domain of exponentials.

Complex Numbers

Complex numbers extend the idea of one-dimensional numbers (real numbers) to two dimensions. This introduction involves the imaginary unit \( i \), where \( i^2 = -1 \). Complex numbers are typically represented as \( a + bi \).

• Here, \( a \) is the real part.
• \( b \) is the imaginary part.

The beauty of complex numbers is in their ability to simplify problems involving periodic phenomena, such as waves. They can be particularly useful in electrical engineering and signal processing.

• Any complex number can be expressed in polar form as \( r(\cos(\theta) + i\sin(\theta)) \), which Euler's formula transforms into \( re^{i\theta} \).
• This representation simplifies multiplication and division of complex numbers.

Understanding complex numbers involves grasping both their algebraic and geometric interpretations, providing a rich area for application in solving equations that are otherwise difficult using only real numbers.

Trigonometric Functions

Trigonometric functions, including sine and cosine, are fundamental in analyzing any periodic or oscillatory processes. Their crucial property is periodicity, meaning these functions repeat their values in regular intervals, called periods.

• The function \( \cos(x) \) repeats every \( 2\pi \) radians, i.e., \( \cos(x + 2\pi k) = \cos(x) \) for any integer \( k \).
• Similarly, \( \sin(x) \) is also periodic with period \( 2\pi \), i.e., \( \sin(x + 2\pi k) = \sin(x) \).

These periodic properties were crucial in simplifying \( \cos\left( \frac{5\pi}{2} \right) \) and \( \sin\left( \frac{5\pi}{2} \right) \) in the original exercise. They illustrate how complex problems can be tackled by translating them into repetitive patterns, simplifying calculations significantly. Understanding the trigonometric functions' behavior provides insight into phenomena such as sound, light waves, and alternating current in electronics.