Problem 39 - Algebra and Trigonometry

Question

In Exercises \(37-44,\) find the product of the complex numbers. Leave answers in polar form. $$ \begin{aligned} &z_{1}=3\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)\\\ &z_{2}=4\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The product \(12(\cos \frac{3\pi}{10} + i \sin \frac{3\pi}{10})\) is the product of complex numbers \(z_1\) and \(z_2\)

Step 1: Identify the given complex numbers

The given complex numbers are \(z_1 = 3(cos(\frac{\pi}{5}) + i \sin(\frac{\pi}{5}))\) and \(z_2 = 4(cos(\frac{\pi}{10}) + i \sin(\frac{\pi}{10}))\). The module and argument of \(z_1\) are 3 and \(\frac{\pi}{5}\) respectively, while for \(z_2\) they are 4 and \(\frac{\pi}{10}\) respectively.

Step 2: Apply the formula for multiplication of complex numbers in polar form

The formula for the product \(z_1 \cdot z_2\) of two complex numbers in polar form is: modulus of \(z_1 . z_2 = |z_1| . |z_2|\) and argument of \(z_1 . z_2 = arg(z_1) + arg(z_2)\). Plugging in our values gives: modulus = \(3 \cdot 4 = 12\), argument = \(\frac{\pi}{5} + \frac{\pi}{10} = \frac{3\pi}{10}\).

Step 3: Write the final result in polar form

The product of our complex numbers \(z_1\) and \(z_2\) in polar form then is: \( z_1 \cdot z_2 = 12(cos(\frac{3\pi}{10}) + i \sin(\frac{3\pi}{10}))\).

Key Concepts

Complex Numbers

Complex numbers are a type of number that extends the concept of one-dimensional numbers to two dimensions. They are composed of a real part and an imaginary part. A complex number is often expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \).

Understanding complex numbers involves grasping how real numbers and imaginary numbers work together on a two-dimensional plane called the complex plane. In this plane:

• The horizontal axis is the real number line.
• The vertical axis is the imaginary number line.

By plotting these numbers, complex numbers can be visually represented and manipulated in practice, making them essential in fields such as engineering and quantum physics.

Multiplication of Complex Numbers

Multiplying complex numbers can be done using the distributive property as it applies to expressions in algebra. However, when complex numbers are represented in polar form (as they are in this exercise), multiplication becomes even more straightforward.

In polar representation, the product of two complex numbers is done by multiplying their moduli (absolute values) and adding their arguments (angles). This is summarized in the formula:

• Modulus of \( z_1 \cdot z_2 \) = \( |z_1| \cdot |z_2| \)
• Argument of \( z_1 \cdot z_2 \) = \( \arg(z_1) + \arg(z_2) \)

This method greatly simplifies computations, especially when dealing with powers of complex numbers, as the rules for exponentiation emerge naturally from repeated multiplication.

Polar Form of Complex Numbers

The polar form of a complex number is a way of expressing complex numbers that can simplify the process of multiplication and division. It expresses a complex number as a magnitude (called the modulus) and an angle (called the argument) derived from the origin in the complex plane.

The polar form of a complex number is represented as:\[ z = r \left( \cos(\theta) + i \sin(\theta) \right) \]where:

• \( r \) is the modulus, representing the distance from the origin to the point \( (a, b) \) in the complex plane.
• \( \theta \) is the argument, the angle from the positive real axis to the line segment connecting the origin to \( (a, b) \).

This form leverages Euler's formula, which links complex exponentials to trigonometric functions, offering a powerful tool for analysis in physics and engineering.

Modulus and Argument

The modulus and argument are two key attributes in the polar representation of complex numbers. They help in understanding the number's position on the complex plane.

**Modulus**
The modulus of a complex number \( z = a + bi \) is its distance from the origin, calculated as the square root of the sum of the squares of its real and imaginary parts:\[ |z| = \sqrt{a^2 + b^2} \]It is a measure of the magnitude of the number.

**Argument**
The argument is the angle formed between the positive real axis and the line connecting the origin to the point \( (a, b) \) on the complex plane. Calculated using the arctangent function, it is expressed in radians:\[ \arg(z) = \tan^{-1} \left( \frac{b}{a} \right) \]Understanding both the modulus and argument is crucial for converting from standard form to polar form, especially when tackling multiplication and division of complex numbers.