Problem 17 - Advanced Engineering Mathematics

Question

$$\begin{aligned}&\left[3 \sqrt{2}\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)\right]\left[10\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right]=30 \sqrt{2}\left[\cos \left(\frac{7 \pi}{4}+\frac{\pi}{3}\right)+i \sin \left(\frac{7 \pi}{4}+\frac{\pi}{3}\right)\right]\\\ &=40.9808+10.9808 i\end{aligned}$$

Step-by-Step Solution

Answer

The product is \(40.9808 + 10.9808i\).

Step 1: Identify the Complex Numbers in Polar Form

Notice that both numbers are in polar form. The first number is \(3\sqrt{2} \left( \cos\frac{7\pi}{4} + i \sin\frac{7\pi}{4} \right)\) and the second number is \(10 \left( \cos\frac{\pi}{3} + i \sin\frac{\pi}{3} \right)\).

Step 2: Multiply the Moduli

Multiply the moduli (magnitudes) of the complex numbers: \(3\sqrt{2}\) and \(10\). Hence, the product of the moduli is: \[3\sqrt{2} \times 10 = 30\sqrt{2}\].

Step 3: Add the Arguments

Add the arguments (angles) of the complex numbers: \(\frac{7\pi}{4}\) and \(\frac{\pi}{3}\). The sum is: \[\frac{7\pi}{4}+\frac{\pi}{3} = \frac{21\pi}{12} + \frac{4\pi}{12} = \frac{25\pi}{12}\].

Step 4: Express the Product in Polar Form

Using the multiplied modulus and added arguments, express the product as: \[30\sqrt{2} \left(\cos \frac{25\pi}{12} + i \sin \frac{25\pi}{12} \right)\].

Step 5: Convert to Rectangular Form

Calculate \(\cos \frac{25\pi}{12}\) and \(\sin \frac{25\pi}{12}\). Evaluate these trigonometric values to convert the polar form into the rectangular form: \[30\sqrt{2}(\cos \frac{25\pi}{12} + i \sin \frac{25\pi}{12}) = 40.9808 + 10.9808i\].

Key Concepts

Polar Form

Complex numbers can be expressed in different forms, and the polar form is one of them. It represents a complex number through a combination of its magnitude (distance from the origin) and angle with respect to the positive real axis. This form is especially useful in various mathematical and engineering fields.

In polar form, a complex number is written as:

• The magnitude or modulus is the distance from the origin to the point, typically represented by \(r\).
• The argument or angle, \(\theta\), shows its direction from the positive real axis, measured in radians.

A complex number \(z\) can be expressed as \(z = r(\cos \theta + i \sin \theta)\), which can also be notated as \(z = r e^{i\theta}\). This form makes it much easier to handle operations like multiplication and division.

Moduli and Argument

The modulus and the argument are key components of a complex number in polar form. They allow us to uniquely identify and manipulate complex numbers.

You can think of the modulus, \(r\), as the 'length' of the vector represented by the complex number. It can be calculated using the formula \(r = \sqrt{a^2 + b^2}\) for a complex number in rectangular form, \(a + bi\).

Moving on to arguments, these are the angles formed by the line representing the complex number with the positive real axis. Angles are measured counterclockwise and usually expressed in radians. When dealing with trigonometry, it's crucial to ensure that the angles (arguments) are added correctly when multiplying complex numbers, as demonstrated in the original exercise.

Trigonometric Functions

Trigonometric functions play a significant role in expressing complex numbers in polar form. These functions help to define the position of the complex number on the complex plane through its angle.

Recall that:

• \(\cos \theta\) gives the x-coordinate (real part)
• \(\sin \theta\) gives the y-coordinate (imaginary part)

When you multiply two complex numbers in polar form, you multiply the moduli and add the arguments. This is because the cosine and sine functions handle the angular positioning on the plane. Understanding these trigonometric properties becomes particularly useful when converting from polar to rectangular form.

Rectangular Form

Rectangular form is another way of expressing complex numbers. Unlike polar form, it directly gives the x and y components (real and imaginary parts), which are visually represented on a cartesian plane.

In rectangular form, a complex number is written as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.

Converting from polar to rectangular involves using trigonometric functions:

• Real part: \(a = r \cos \theta\)
• Imaginary part: \(b = r \sin \theta\)

By substituting these back, we go from complex expressions like \(30\sqrt{2}(\cos \frac{25\pi}{12} + i \sin \frac{25\pi}{12})\) to something more straightforward, \(40.9808 + 10.9808i\), which offers a clearer visual understanding of the complex number on the plane.