Problem 5 - Advanced Engineering Mathematics

Question

$$\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to 1 + i.

Step 1: Understand the Given Expression

The expression provided is \( \sqrt{2}\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right) \). This is in the form of a complex number expressed in polar form, where \( r = \sqrt{2} \) and \( \theta = \frac{\pi}{4} \).

Step 2: Convert to Rectangular Form

We convert the polar form \( r(\cos \theta + i \sin \theta) \) to rectangular form \( a + bi \). Calculate \( a \) and \( b \) as follows: - For the real part \( a = r \cos \theta = \sqrt{2} \cos \frac{\pi}{4} \)- For the imaginary part \( b = r \sin \theta = \sqrt{2} \sin \frac{\pi}{4} \)

Step 3: Evaluate Trigonometric Functions

Evaluate the trigonometric functions \( \cos \frac{\pi}{4} \) and \( \sin \frac{\pi}{4} \). Both are equal to \( \frac{\sqrt{2}}{2} \).

Step 4: Calculate Real and Imaginary Parts

Use the values from Step 3 to compute:- \( a = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1 \)- \( b = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1 \)

Step 5: Write the Result in Rectangular Form

Combine the real and imaginary parts to express the final result in rectangular form: \( 1 + i \).

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be represented in different forms, and the polar form is one of the most intuitive for understanding their geometric nature. In polar form, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the argument or angle from the positive x-axis. Polar form is particularly useful for operations like multiplication and division of complex numbers. It is also very handy when dealing with trigonometric functions, as it neatly combines them. The magnitude \( r \) is the distance from the origin to the point in the complex plane, often calculated as \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts of the complex number in rectangular form. The angle \( \theta \) indicates the direction of the vector representing the complex number.

Rectangular Form of Complex Numbers

The rectangular form, also called the Cartesian form, represents a complex number as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This form is very useful for straightforward addition and subtraction of complex numbers and is typically the starting point for introducing complex numbers in mathematics courses. When talking about the rectangular form, we think about the complex number as a point or vector with coordinates \( (a, b) \) in a two-dimensional plane, known as the complex plane. Transitioning from polar to rectangular form is useful for performing tasks that require explicit computation of the real and imaginary components, as evidenced in the given exercise.

Trigonometric Functions

Trigonometric functions, specifically cosine and sine, are essential when working with complex numbers in polar form. They relate angles to the sides of a right triangle and appear in the polar form of complex numbers. Here, \( \cos \theta \) gives the horizontal distance (or the real part when converting to rectangular form) and \( \sin \theta \) provides the vertical distance (or the imaginary part). For example, in the exercise, \( \cos \frac{\pi}{4} \) and \( \sin \frac{\pi}{4} \) both equal \( \frac{\sqrt{2}}{2} \). These specific values arise from the angle \( \frac{\pi}{4} \), equivalent to 45 degrees, where the complexities and symmetries of the unit circle make calculations straightforward with these functions.

Mathematical Conversion Techniques

Converting complex numbers between polar and rectangular forms involves a few steps and is based on understanding the relationship between angle and radius in polar form and the real and imaginary components in rectangular form. The process of conversion involves calculating the real part `a` as \( r \cos \theta \) and the imaginary part `b` as \( r \sin \theta \). This technique is crucial because it translates the geometric representation of a complex number into a more algebraically manageable form, allowing for various mathematical operations. Efficient conversion techniques are foundational in tackling complex problems in engineering and physics, as it simplifies handling of rotational and oscillatory systems. In essence, recognizing and mastering these conversion steps enables powerful leverage over mathematical expressions.