Problem 10 - Advanced Engineering Mathematics

Question

$$e^{-0.23-i}=e^{-0.23} \cos (-1)+i e^{-0.23} \sin (-1)=0.4293-0.6686 i$$

Step-by-Step Solution

Answer

The expression evaluates to \( 0.4293 - 0.6686i \).

Step 1: Rewrite in Exponential Form

First, we need to understand that the given expression can be rewritten using the complex exponential form. The expression is given as: \[ e^{-0.23 - i} = e^{(-0.23)} imes e^{-i} \]This separates the real and imaginary components in the exponent.

Step 2: Separate Real and Imaginary Parts

Next, we express each component separately:- The real part is \( e^{-0.23} \)- The imaginary part \( e^{-i} \) can be expressed using Euler's formula as \( \cos(-1) + i \sin(-1) \).

Step 3: Calculate Exponential Part

Calculate \( e^{-0.23} \):\[ e^{-0.23} \approx 0.7945 \]This value will be used to scale the cosine and sine terms from the Euler's formula.

Step 4: Evaluate Cosine and Sine

Now, calculate the cosine and sine:- \( \cos(-1) \approx 0.5403 \)- \( \sin(-1) \approx -0.8415 \)These values are derived from the unit circle.

Step 5: Multiply to Obtain Real and Imaginary Parts

Multiply the calculated exponential part by the cosine and sine to find the real and imaginary components:\[ e^{-0.23} imes \cos(-1) \approx 0.7945 \times 0.5403 \approx 0.4293 \]\[ e^{-0.23} imes \sin(-1) \approx 0.7945 \times (-0.8415) \approx -0.6686 \]

Step 6: Combine Results

Combine the real and imaginary components to present the complex number:\[ e^{-0.23 - i} = 0.4293 - 0.6686i \]

Key Concepts

Euler's Formula

Euler's Formula is a fascinating link between trigonometry and complex numbers. It expresses complex exponential functions in terms of trigonometric functions. The formula is given by: \[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]where \( i \) is the imaginary unit and \( \theta \) is a real number. This formula helps us express complex numbers in a compact and insightful way.

• \( \cos(\theta) \) represents the real part of the complex number.
• \( i\sin(\theta) \) represents the imaginary part with \( i \) indicating a 90-degree rotation on the complex plane.

This connection is particularly useful when working with complex numbers in polar form. In the given exercise, \(-1\) in the exponent corresponds to \( \theta \) in Euler's Formula, enabling us to easily evaluate cosine and sine functions.

Real and Imaginary Components

Complex numbers are composed of two parts: real and imaginary components. These can be visualized on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Every complex number can be expressed in the form:\[ a + bi \]where \( a \) is the real part and \( b \) is the imaginary part.

• The real part, \( a \), is the component that aligns with the real axis.
• The imaginary part, \( bi \), aligns with the imaginary axis and is multiplied by the imaginary unit \( i \).

In the original exercise, we found the real component as \( 0.4293 \) and the imaginary component as \(-0.6686i\). These values were derived using Euler's Formula and help represent the point \( 0.4293 - 0.6686i \) on the complex plane.

Exponential Form of Complex Numbers

The exponential form of complex numbers provides a powerful way to manipulate and understand these numbers. It combines both the real and imaginary components in an elegant expression, making calculations more streamlined. The general form is:\[ re^{i\theta} \]Here, \( r \) represents the magnitude (or modulus) and \( \theta \) is the argument (or angle) of the complex number.

• The magnitude \( r \) is equivalent to the length of the vector representing the complex number from the origin.
• The argument \( \theta \) is the angle between this vector and the positive real axis.

For the exercise, the expression \( e^{-0.23 - i} \) is transformed using exponential form, where \(-0.23\) is the real part of the exponent affecting the magnitude, and \(-1\) corresponds to the argument as used in Euler's Formula. This form simplifies multiplication and division, especially in complex number operations.