Problem 29 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ 2 \sqrt{3}-2 i $$

Step-by-Step Solution

Answer

The polar form is \(4(\cos(\frac{11\pi}{6}) + i \sin(\frac{11\pi}{6}))\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(2\sqrt{3} - 2i\). Identify the real part \(a\) and the imaginary part \(b\). Here, \(a = 2\sqrt{3}\) and \(b = -2\).

Step 2: Calculate the Magnitude \(r\)

The magnitude \(r\) of a complex number \(a + bi\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\). Substitute the values to find \(r\). \[r = \sqrt{(2\sqrt{3})^2 + (-2)^2} = \sqrt{4 \times 3 + 4} = \sqrt{12 + 4} = \sqrt{16} = 4\]

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

The argument \(\theta\) is found using \(\tan(\theta) = \frac{b}{a}\). Since \(a = 2\sqrt{3}\) and \(b = -2\), calculate \(\tan(\theta)\).\[ \tan(\theta) = \frac{-2}{2\sqrt{3}} = -\frac{1}{\sqrt{3}}\]This corresponds to \(\theta = -\frac{\pi}{6}\) or equivalently, \(\theta = \frac{11\pi}{6}\) after adjusting to the interval \([0, 2\pi)\).

Step 4: Write the Complex Number in Polar Form

Using the calculated magnitude \(r\) and the argument \(\theta\), write the complex number in polar form as follows:\[2\sqrt{3} - 2i = 4 \left( \cos\left(\frac{11\pi}{6}\right) + i \sin\left(\frac{11\pi}{6}\right) \right)\]

Key Concepts

Polar Form

Complex numbers can be expressed in two distinct forms: rectangular (or Cartesian) and polar. In rectangular form, a complex number is written as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part involving \(i\), the imaginary unit. Polar form, on the other hand, represents a complex number in terms of magnitude \(r\) and angle \(\theta\). This approach offers a geometric perspective, capturing the position of the number in the complex plane. To convert from rectangular to polar form, use the formula:

• Magnitude \(r = \sqrt{a^2 + b^2}\)
• Argument \(\theta\) is such that \(\tan(\theta) = \frac{b}{a}\)

This is often expressed as \(r(\cos \theta + i \sin \theta)\). The advantage of polar form is its simplicity when multiplying, dividing, or finding powers and roots of complex numbers. It unveils a new way to handle complex arithmetic by focusing on angles and distances in a circle.

Magnitude of a Complex Number

The magnitude of a complex number, denoted by \(r\), is akin to the length or absolute value in the real number system. It measures how far the point is from the origin in the complex plane.
The formula for finding the magnitude is \(r = \sqrt{a^2 + b^2}\), where \(a\) is the real part and \(b\) is the imaginary part.
In the context of the exercise, the given complex number is \(2\sqrt{3} - 2i\). Calculating its magnitude involves squaring both parts and taking the square root

• \((2\sqrt{3})^2 = 4 \times 3 = 12\)
• \((-2)^2 = 4\)

Adding these results and taking the square root gives \(\sqrt{16} = 4\).
This tells us the distance of the number from the origin is 4 units in the complex plane.

Argument of a Complex Number

The argument of a complex number, \(\theta\), represents the angle between the positive real axis and a line joining the number to the origin in the complex plane.
For the complex number \(a + bi\), the argument can be found using the arctangent function: \(\tan(\theta) = \frac{b}{a}\).
In our example, with \(a = 2\sqrt{3}\) and \(b = -2\), we calculate:

• \(\tan(\theta) = \frac{-2}{2\sqrt{3}} = -\frac{1}{\sqrt{3}}\)

Analyzing this ratio, we look for an angle whose tangent matches this value, resulting in \(\theta = -\frac{\pi}{6}\).
However, because angle measures in polar form range from 0 to \(2\pi\), this value is adjusted to a positive angle: \(\theta = \frac{11\pi}{6}\).
This reflects a full rotation from the positive x-axis, which places the vector in the fourth quadrant of the Cartesian system.

Imaginary Numbers

Imaginary numbers are a fundamental component of complex numbers, represented by the symbol \(i\), which is defined as the square root of -1.
This imaginary unit allows mathematicians to explore solutions to equations lacking real number answers, such as \(x^2 + 1 = 0\).
In the equation, \(i\) cannot be ignored because it broadens the number system to incorporate imaginary and complex numbers.

• \(a + bi\) formulates a complex number where \(a\) is the real component, and \(b\) is the imaginary component.
• The imaginary part, \(bi\), is distinguished by \(b\) being a real number multiplied by \(i\).

Complex numbers can be visualized in the complex plane, where the horizontal axis represents real numbers and the vertical one represents imaginary numbers.
This perspective allows for a more comprehensive manipulation and a deeper understanding of the dynamic nature of numbers.