Problem 18 - Advanced Engineering Mathematics

Question

$$\left[4 \sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right]\left[\sqrt{2}\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right]=8\left[\cos \left(\frac{\pi}{4}+\frac{3 \pi}{4}\right)+i \sin \left(\frac{\pi}{4}+\frac{3 \pi}{4}\right)\right]=-8$$

Step-by-Step Solution

Answer

-8

Step 1: Understand the Given Expression

We have the expression \( \left[4 \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\right]\left[\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\right] \) which is in polar form for complex numbers, taking advantage of Euler's formula \( e^{i\theta} = \cos \theta + i\sin \theta \).

Step 2: Multiply Moduli

In polar form, multiplying two complex numbers involves multiplying their moduli: \( 4\sqrt{2} \times \sqrt{2} = 4 \times 2 = 8 \).

Step 3: Add Angles

For the angle part, we add the angles of the two complex numbers: \( \frac{\pi}{4} + \frac{3\pi}{4} = \pi \).

Step 4: Reconstruct the Product

Now, reconstruct the product as \( 8 \left( \cos \pi + i \sin \pi \right) \). Known trigonometric values are \( \cos \pi = -1 \) and \( \sin \pi = 0 \).

Step 5: Simplify the Expression

Substitute the trigonometric values: \( 8 \times \left( -1 + 0i \right) = -8 \).

Step 6: Conclude the Solution

The final result of the given expression is \(-8\), as the operations confirm this value as stated in the problem.

Key Concepts

Euler's formula

Euler's formula is a cornerstone in understanding complex numbers and their behavior. It establishes a profound connection between trigonometry and complex exponential functions. This formula states that for any real number \( \theta \), the equation \( e^{i\theta} = \cos \theta + i \sin \theta \) holds true. Here's what each part means:

• \( e^{i\theta} \) can be understood as a rotation in the complex plane by an angle \( \theta \).
• \( \cos \theta \) and \( \sin \theta \) represent the projections onto the real and imaginary axes of a circle of radius 1.
• The imaginary unit \( i \) signifies a 90-degree rotation, so \( i \sin \theta \) adds the imaginary component.

Applying Euler's formula allows us to simplify complex expressions, especially when combined with angles, by visualizing them as rotations and utilizing exponential rules. This foundational tool is essential when working with complex numbers in polar form or when performing multiplication.

Polar Form

The polar form of a complex number provides an alternative way to express it, focusing on its magnitude and direction. Given a complex number \( z \), it can be written as \( r ( \cos \theta + i \sin \theta ) \) or alternatively as \( r e^{i\theta} \), thanks to Euler's formula. Here's how to understand it:

• \( r \) is the modulus of the complex number, indicating its distance from the origin in the complex plane.
• \( \theta \) is the argument or angle, showing the direction of the line connecting the origin to the point in the plane.

Expressing a complex number in polar form highlights its geometric properties and simplifies operations like multiplication and division. It also directly relates to Euler's formula, making complex analysis more intuitive and less cumbersome. This form is especially useful when dealing with powers and roots of complex numbers.

Trigonometric Values

Understanding trigonometric values is crucial when working with complex numbers in both Cartesian and polar forms. Key angles like \( \frac{\pi}{4} \) and \( \pi \) appear often due to their simplicity and their significant roles in geometry. To break it down:

• \( \frac{\pi}{4} \) corresponds to a 45-degree angle, where \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). These values are equal because the angle lies along the angle bisector in the unit circle's first quadrant.
• \( \pi \) or 180 degrees represents a point directly on the negative real axis, leading to \( \cos \pi = -1 \) and \( \sin \pi = 0 \). This tells us that the associated vector is pointing in the negative x-direction.

Recognizing these trigonometric values at key angles helps simplify complex number calculations. It reduces errors in operations, especially multiplication, and aids in understanding deeper relationships within mathematical models.

Multiplication of Complex Numbers

Multiplying complex numbers becomes much simpler when they are in polar form. Here's a step-by-step on how it works:

• First, multiply their moduli (the magnitudes of the vectors representing the complex numbers). For example, if \( |z_1| = a \) and \( |z_2| = b \), then the modulus of the product is \( a \times b \).
• Next, add their angles (arguments). If \( z_1 \) has an angle \( \alpha \) and \( z_2 \) has an angle \( \beta \), the resulting angle is \( \alpha + \beta \).
• The product is then given by \( (a \cdot b)( \cos(\alpha + \beta) + i \sin(\alpha + \beta) ) \).

This process simplifies what would be a lengthy multiplication if the numbers were in their standard form. By converting to polar form, operations reduce to simple arithmetic manipulations of scalars and angles, allowing you to arrive at the result swiftly and accurately. Such elegance is a key reason why engineers and scientists often prefer using the polar form in practical applications.