Problem 33 - Algebra and Trigonometry

Question

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

Polar form is \( 4 \text{cis} \frac{11\pi}{6} \).

Step 1: Identify Real and Imaginary Parts

The given complex number is in the form \( a + bi \). For the number \( 2\sqrt{3} - 2i \), \( a = 2\sqrt{3} \) and \( b = -2 \).

Step 2: Calculate Magnitude (r)

The magnitude \( r \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \). Substituting the values we have:\[ r = \sqrt{(2\sqrt{3})^2 + (-2)^2} = \sqrt{4 \cdot 3 + 4} = \sqrt{12 + 4} = \sqrt{16} = 4 \]

Step 3: Find Argument (θ)

The argument \( \theta \) can be found using \( \theta = \tan^{-1}(\frac{b}{a}) \). Here, \( \tan^{-1}(\frac{-2}{2\sqrt{3}}) = \tan^{-1}(-\frac{1}{\sqrt{3}}) \). \( \tan^{-1}(-\frac{1}{\sqrt{3}}) \) corresponds to \( \theta = -\frac{\pi}{6} \) in the fourth quadrant. Since we need \( \theta \) not negative, add \( 2\pi \) to bring it within the range \( 0 \) to \( 2\pi \):\[ \theta = 2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6} \]

Step 4: Write in Polar Form

The polar form of a complex number is \( r(\cos(\theta) + i\sin(\theta)) \). Substituting \( r = 4 \) and \( \theta = \frac{11\pi}{6} \):\[ 4(\cos(\frac{11\pi}{6}) + i\sin(\frac{11\pi}{6})) \]. Therefore, the polar form is \( 4 \text{cis} \frac{11\pi}{6} \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. They are usually written in the form \( a + bi \), where:

• \( a \) is the real part
• \( b \, (i) \) is the imaginary part, where \( i \) is the imaginary unit with the property \( i^2 = -1 \)

For example, in the complex number \( 2\sqrt{3} - 2i \), the real part is \( 2\sqrt{3} \), and the imaginary part is \( -2i \). Dealing with complex numbers enables us to solve quadratic equations that do not have real solutions, expanding the set of numbers we can work with. They are crucial in various fields like engineering and physics for representing phenomena such as electrical circuits and wave patterns.

Magnitude and Argument

The magnitude of a complex number is a measure of its distance from the origin in the complex plane and is denoted by \( r \). It can be calculated using the formula \( r = \sqrt{a^2 + b^2} \). In our example, for the complex number \( 2\sqrt{3} - 2i \), the magnitude calculates to \( 4 \). The concept of magnitude is similar to the length of a vector in two dimensions.

The argument of a complex number is the angle \( \theta \) that the line representing the complex number makes with the positive x-axis in a counterclockwise direction. It can be calculated using \( \theta = \tan^{-1}(\frac{b}{a}) \). In this exercise, the initial calculation results in a negative angle because it is positioned in the fourth quadrant of the complex plane. We modify it to lie within a positive range by adding \( 2\pi \), resulting in \( \theta = \frac{11\pi}{6} \). Knowing both the magnitude and the argument allows us to express a complex number in polar form.

Real and Imaginary Parts

The real and imaginary parts are fundamental aspects of complex numbers. They define the coordinates of a point in the complex plane.

• The real part, \( a \), is the x-coordinate, signifying the horizontal position of the number.
• The imaginary part, \( b \), determines the vertical position and is often represented with the imaginary unit \( i \), making it the y-coordinate.

Together, these components determine a complex number's position on the plane. By transforming these coordinates into magnitude (distance from the origin) and argument (angle from the x-axis), we switch from Cartesian to polar coordinates. This transformation simplifies multiplying and dividing complex numbers, since we can multiply magnitudes and add arguments rather than performing operations on the real and imaginary parts directly.