Problem 39 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$4 \sqrt{3}-4 i$$

Step-by-Step Solution

Answer

The complex number in polar form is \(8 \left( \cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6} \right)\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(4\sqrt{3} - 4i\). This means the real part \(a = 4\sqrt{3}\) and the imaginary part \(b = -4\).

Step 2: Calculate the Modulus

The modulus \(r\) of the complex number is calculated using the formula \(r = \sqrt{a^2 + b^2}\). Substitute the values of \(a\) and \(b\): \[ r = \sqrt{(4\sqrt{3})^2 + (-4)^2} = \sqrt{48 + 16} = \sqrt{64} = 8 \]

Step 3: Determine the Argument

The argument \(\theta\) is found using the formula \(\theta = \arctan\left(\frac{b}{a}\right)\). Substitute the values of \(a\) and \(b\): \[ \theta = \arctan\left(\frac{-4}{4\sqrt{3}}\right) = \arctan\left(\frac{-1}{\sqrt{3}}\right) \]The angle corresponds to the standard angle \(\theta = \frac{11\pi}{6}\) where the complex number is in the fourth quadrant.

Step 4: Write in Polar Form

With the modulus \(r = 8\) and the argument \(\theta = \frac{11\pi}{6}\), the polar form of the complex number is written as:\[ 8 \left( \cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6} \right) \]

Key Concepts

Complex Numbers

A complex number is a type of number that has both a real and an imaginary part. The real part is a regular number, like 5 or -3, and the imaginary part involves the square root of a negative number, denoted by 'i'. For example, in the complex number \(4\sqrt{3} - 4i\), \(4\sqrt{3}\) is the real part, and \(-4i\) is the imaginary part.
Understanding complex numbers is essential because they allow us to expand the rules of arithmetic to solve equations that don't have solutions in real numbers alone. They are often represented as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

• The component \(i\) is defined as \(\sqrt{-1}\).
• Complex numbers can be graphed on a plane, with the real part on the x-axis and the imaginary part on the y-axis.
• They are useful in engineering, physics, and advanced mathematics.

Modulus

The modulus of a complex number is akin to its distance from the origin on the complex plane. For a complex number \(a + bi\), the modulus \(r\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\). For example, if the complex number is \(4\sqrt{3} - 4i\), the modulus turns out to be 8.
Calculating the modulus is crucial for converting a complex number to polar form, which makes operations like multiplication and division more manageable.

• The modulus serves as the radius of a circle in the polar coordinate representation.
• It can never be negative, as it represents a distance.
• In polar form, the number is expressed as \(r\left(\cos\theta + i\sin\theta\right)\).

Understanding the modulus helps us visualize and manipulate complex numbers much like plotting points on a two-dimensional graph.

Argument

The argument of a complex number is the angle \(\theta\) the number makes with the positive real axis in the complex plane. It is calculated using the formula \(\theta = \arctan\left(\frac{b}{a}\right)\), where \(a\) is the real part and \(b\) is the imaginary part.
For the complex number \(4\sqrt{3} - 4i\), the argument is \(\frac{11\pi}{6}\). The argument helps specify the direction from the origin where the complex number is located on the complex plane.

• The argument can range from \(0\) to \(2\pi\) radians.
• It explains how the complex number relates to the real axis.
• Positive arguments rotate counterclockwise, while negative ones go clockwise.

Using the argument alongside the modulus allows us to completely describe complex numbers in polar form, which simplifies computations involving powers and roots.