Problem 36 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ 8 i $$

Step-by-Step Solution

Answer

The polar form is \( 8(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) \).

Step 1: Identify the Real and Imaginary Parts

The given complex number is just the imaginary part, which is \(8i\). Thus, the real part \(x = 0\) and the imaginary part \(y = 8\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(x + yi\) is given by the formula \( r = \sqrt{x^2 + y^2} \). Substituting the values, we get \( r = \sqrt{0^2 + 8^2} = \sqrt{64} = 8 \).

Step 3: Determine the Argument \(\theta\)

Since the real part \(x = 0\) and the imaginary part \(y = 8\) is positive, the complex number lies on the positive imaginary axis. In this case, \( \theta = \frac{\pi}{2} \).

Step 4: Write the Polar Form

Having found both the magnitude and the argument, the polar form of the complex number is given by \( 8 \left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right) \).

Key Concepts

Polar Form

Complex numbers can be represented in different forms. One of these forms is the polar form. This representation is great for understanding the geometric properties of complex numbers. Instead of expressing complex numbers in their rectangular form, which is usually as \(x + yi\), the polar form uses the magnitude of the number and an angle known as the argument.

The polar form of a complex number is given by:

• \( r(\cos \theta + i \sin \theta) \)
• The parameter \( r \) is the magnitude of the complex number.
• The angle \( \theta \) is the argument.

Introducing the polar form helps to simplify many operations like multiplication and division of complex numbers.

Magnitude of Complex Numbers

The magnitude of a complex number is a measure of its size or length from the origin in the complex plane. For a complex number \(x + yi\), the magnitude is calculated using the formula:

• \[ r = \sqrt{x^2 + y^2} \]

This formula is derived from the Pythagorean theorem, treating the real and imaginary parts as the two legs of a right triangle.

In the context of our example, for \(8i\):

• The real part is 0, so \(x = 0\).
• The imaginary part is 8, so \(y = 8\).
• Magnitude is \( r = \sqrt{0^2 + 8^2} = 8 \).

Therefore, the magnitude is simply the distance from the origin to the point on the imaginary axis where our complex number lies.

Argument of Complex Numbers

The argument of a complex number is the angle formed with the positive real axis in the complex plane. It's a way to express the direction of the complex number.

For \(x + yi\), to find the argument \(\theta\):

• Use \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).

However, when the real part \(x=0\), as in the example of \(8i\), the complex number is located directly on the imaginary axis. This makes the argument equal to:

• \(\theta = \frac{\pi}{2}\) or \(90^\circ\).

This indicates that the number is straight up on the imaginary axis.

Imaginary Part

The imaginary part of a complex number is the coefficient of \(i\), which is the unit imaginary number defined as \(i^2 = -1\). In a complex number form \(x + yi\), \(y\) is known as the imaginary part.

Consider the complex number \(8i\):

• The real part \(x = 0\), which means there is no horizontal component.
• The imaginary part is \(y = 8\), so it simply represents the vertical component on the imaginary plane.

Understanding the imaginary part is essential because it defines how far up or down a complex number is on the vertical axis. In operations with complex numbers, the imaginary part plays a crucial role in determining outcomes in calculations involving addition, subtraction, and more intricate operations like finding roots or exponents.