Problem 47 - Algebra and Trigonometry

Question

In Exercises \(45-52,\) find the quotient \(\frac{z_{1}}{z_{2}}\) of the complex numbers. Leave answers in polar form. In Exercises \(49-50,\) express the argument as an angle between \(0^{\circ}\) and \(360^{\circ}\). $$ \begin{aligned} &z_{1}=3\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)\\\ &z_{2}=4\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The quotient of the complex numbers \(z_{1}\) and \(z_{2}\) is \(\frac{3}{4}\left(\cos\left(\frac{\pi}{5}-\frac{\pi}{10}\right)+i \sin\left(\frac{\pi}{5}-\frac{\pi}{10}\right)\right)\)

Step 1: Express complex numbers in polar form

The complex numbers are already expressed in their polar forms: \(z_{1}=3\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)\) and \(z_{2}=4\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)\)

Step 2: Calculate the magnitude and argument of the quotient

Divide the magnitudes (3/4) and subtract the angles \(\left(\frac{\pi}{5}-\frac{\pi}{10}\right)\) to find the magnitude and argument of the quotient.

Step 3: Express the quotient in polar form

The quotient in polar form would be: \(\frac{3}{4}\left(\cos\left(\frac{\pi}{5}-\frac{\pi}{10}\right)+i \sin\left(\frac{\pi}{5}-\frac{\pi}{10}\right)\right)\)

Key Concepts

Polar Form

Complex numbers can be represented in a few different ways, but one of the most beneficial is polar form. This representation leverages the magnitude and the angle (known as the argument) to describe a complex number. In polar form, a complex number is expressed as:

• \( z = r(\cos \theta + i \sin \theta) \)

Where:

• \( r \) represents the magnitude of the complex number.
• \( \theta \) is the argument, encapsulating the direction/angle.

In the original exercise, both given complex numbers \( z_1 \) and \( z_2 \) are already expressed in polar form. This makes calculations involving addition, subtraction, or division simpler as compared to traditional Cartesian forms. Dividing two complex numbers in polar form is straightforward:

• Divide their magnitudes.
• Subtract their arguments.

This simplicity allows for easier computation during exercises involving complex numbers.

Magnitude of Complex Numbers

The magnitude, also known as the modulus of a complex number, measures how 'big' the complex number is. You can imagine this as the length of the vector representing the complex number on a graph.
For a complex number given in polar form \( z = r(\cos \theta + i \sin \theta) \), the magnitude is represented by \( r \).
Finding the magnitude doesn't depend on the angle; it purely considers the size of the number.

• For \( z_1 \), the magnitude is 3.
• For \( z_2 \), the magnitude is 4.
When dealing with division, as seen in the problem, you simply divide the magnitudes. For the quotient \( \frac{z_1}{z_2} \), the magnitude becomes \( \frac{3}{4} \). This simple operation illustrates why polar form makes such problems easier to handle. It abstracts away the real and imaginary complexities.

Argument of Complex Numbers

The argument of a complex number is the angle it forms with the positive real axis in the complex plane. Represented in polar form as \( \theta \), it reveals the direction in which the complex number lies.
Calculating the argument involves applying the arctan function to the ratio of the imaginary to the real part when in Cartesian form, or using the given value in polar form.
For the given problem:

• \( z_1 \) has an argument \( \theta_1 = \frac{\pi}{5} \).
• \( z_2 \) has an argument \( \theta_2 = \frac{\pi}{10} \).

Subtracting these arguments when dividing the complex numbers gives us the argument of the resultant quotient:

• \( \theta_{quotient} = \frac{\pi}{5} - \frac{\pi}{10} = \frac{\pi}{10} \).

This calculation shows how the angle of the result relates to those of the initial numbers. It's crucial for translating complex numbers into their geometric representation, allowing for deeper understanding of their properties and behaviors.