Problem 9 - Advanced Engineering Mathematics

Question

Express \(e^{z}\) in the form \(a+i b\). \(z=5 i\)

Step-by-Step Solution

Answer

The expression is approximately \(0.28366 - 0.95892i\).

Step 1: Identify the given expression

We are given the expression of the form \(e^{z}\) where \(z = 5i\). This means \(z\) is purely imaginary.

Step 2: Use Euler's Formula

Recall Euler's formula: \(e^{ix} = \cos(x) + i\sin(x)\). Here, we substitute \(x = 5\), resulting in \(e^{5i} = \cos(5) + i\sin(5)\).

Step 3: Calculate \(\cos(5)\) and \(\sin(5)\)

Using a calculator or trigonometric table, determine \(\cos(5)\) and \(\sin(5)\) where the angle is in radians:- \(\cos(5) \approx 0.28366\)- \(\sin(5) \approx -0.95892\)

Step 4: Combine results into a complex number

Substitute the calculated cosine and sine into the expression:\(e^{5i} = 0.28366 - 0.95892i\). Thus, the expression in the form \(a + ib\) is \(0.28366 - 0.95892i\).

Key Concepts

Euler's Formula

Euler's formula is a fundamental bridge in mathematics, connecting complex numbers and trigonometry. It's represented as:

• \( e^{ix} = \cos(x) + i \sin(x) \)

Here, \( i \) is the imaginary unit, and \( x \) is a real number (often an angle in radians). This formula elegantly represents the exponential form of a complex number using trigonometric functions, showing how they cycle through different values.

The beauty of Euler’s formula lies in its ability to express complex exponentiation in terms of familiar sine and cosine functions. For example, when we have \( e^{5i} \), it translates into \( \cos(5) + i \sin(5) \). These functions allow us to convert an expression from exponential to rectangular form, providing a more intuitive view of complex numbers.

Imaginary Number

Imaginary numbers are a crucial part of complex numbers. They are called 'imaginary' simply because they extend beyond the real line of numbers we're commonly used to. The unit of imaginary numbers is denoted by \( i \), where:

• \( i^2 = -1 \)

By definition, an imaginary number is any real number multiplied by \( i \). For example, in the expression \( 5i \), the number is purely imaginary because it doesn’t contain a real part.

These numbers are essential in various fields of science and engineering, providing solutions that real numbers alone cannot. Imaginary numbers enable us to compute square roots of negative numbers, among other things, which is impossible with just real numbers. This creates a complete field called complex numbers, encompassing both real and imaginary parts.

Polar Form of Complex Numbers

The polar form of complex numbers offers a unique way to express complex numbers in terms of magnitude and angle, rather than just real and imaginary components. This form is quite useful for multiplication and division of complex numbers.

• The general representation is \( r(\cos\theta + i \sin\theta) \)
• Here, \( r \) is the modulus or magnitude of the complex number
• \( \theta \) is the angle or argument of the complex number

For example, a complex number \( z = a + ib \) can be translated into polar form using the relationships:

• \( r = \sqrt{a^2 + b^2} \)
• \( \theta = \tan^{-1}(\frac{b}{a}) \)

This transformation allows for more manageable operations with complex numbers. The polar form highlights the geometric interpretation of complex numbers as points or vectors in a plane, showing how they rotate and scale, providing profound insights into their behavior.