Problem 53 - Precalculus Mathematics for Calculus

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\). Express your answer in polar form. $$z_{1}=\cos \pi+i \sin \pi, \quad z_{2}=\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}$$

Step-by-Step Solution

Answer

Product: \(e^{i\frac{4\pi}{3}}\), Quotient: \(e^{i\frac{2\pi}{3}}\).

Step 1: Identify the given complex numbers in polar form

The complex number \(z_1 = \cos \pi + i \sin \pi\) is in polar form, written as \(e^{i\pi}\). Similarly, \(z_2 = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3}\) is written as \(e^{i\frac{\pi}{3}}\).

Step 2: Find the product of the complex numbers

Using the property of multiplying complex numbers in polar form \(e^{i\theta_1} e^{i\theta_2} = e^{i(\theta_1+\theta_2)}\), the product \(z_1 z_2 = e^{i\pi} e^{i\frac{\pi}{3}} = e^{i(\pi + \frac{\pi}{3})}\). Simplifying, \(\pi + \frac{\pi}{3} = \frac{4\pi}{3}\), so \(z_1 z_2 = e^{i\frac{4\pi}{3}}\).

Step 3: Find the quotient of the complex numbers

Using the property of dividing complex numbers in polar form \(\frac{e^{i\theta_1}}{e^{i\theta_2}} = e^{i(\theta_1-\theta_2)}\), the quotient \(\frac{z_1}{z_2} = \frac{e^{i\pi}}{e^{i\frac{\pi}{3}}} = e^{i(\pi - \frac{\pi}{3})}\). Simplifying, \(\pi - \frac{\pi}{3} = \frac{2\pi}{3}\), so \(\frac{z_1}{z_2} = e^{i\frac{2\pi}{3}}\).

Step 4: Express both results in polar form

The product is \(z_1 z_2 = e^{i\frac{4\pi}{3}}\) and the quotient is \(\frac{z_1}{z_2} = e^{i\frac{2\pi}{3}}\). This express both solutions neatly in polar form as specified in the problem statement.

Key Concepts

Polar Coordinates

Complex numbers can be represented in several forms, and one popular method is using polar coordinates.
Polar form expresses a complex number in terms of its magnitude (or modulus) and angle (or argument).
A complex number in polar form is given by \( z = r (\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle with the positive x-axis (real axis).
This can also be represented using Euler's formula as \( z = re^{i\theta} \).
This form is particularly useful when dealing with multiplication and division of complex numbers.

• Magnitude (\( r \)): It is the distance from the origin to the point in the complex plane and can be calculated using \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts, respectively.
• Argument (\( \theta \)): This is the angle formed with the positive x-axis and can be determined using trigonometric functions such as \( \tan^{-1} (b/a) \).

Polar coordinates provide a powerful way to visualize and manipulate complex numbers. This helps simplify operations like multiplication and division.

Multiplication of Complex Numbers

When multiplying complex numbers in polar form, the process is straightforward and intuitive.
Given two complex numbers \( z_1 = r_1 e^{i\theta_1} \) and \( z_2 = r_2 e^{i\theta_2} \), the product can be easily calculated using the formula:

• \( z_1 z_2 = (r_1 r_2) e^{i(\theta_1 + \theta_2)} \)

This means:

• Multiply the magnitudes: \( r_1 \times r_2 \).
• Add the angles: \( \theta_1 + \theta_2 \).

This method effectively combines and compacts the information of both complex numbers under multiplication.
The simplicity of adding angles and multiplying moduli makes it a preferred method compared to multiplication in rectangular form.
In the given exercise, we used this property to find the product of two complex numbers, resulting in polar form \( e^{i\frac{4\pi}{3}} \). This is both efficient and clean in terms of arithmetic computation.

Division of Complex Numbers

Dividing complex numbers in polar form follows a method analogous to multiplication but generally seen as even easier.
For two complex numbers \( z_1 = r_1 e^{i\theta_1} \) and \( z_2 = r_2 e^{i\theta_2} \), the division is calculated by:

• \( \frac{z_1}{z_2} = \left(\frac{r_1}{r_2}\right) e^{i(\theta_1 - \theta_2)} \)

This involves:

• Dividing the magnitudes: \( \frac{r_1}{r_2} \).
• Subtracting the angles: \( \theta_1 - \theta_2 \).

This operation is particularly straightforward because the operations are independent - magnitudes and angles are managed separately.
In the problem at hand, using this formula provided a neat solution for the quotient as \( e^{i\frac{2\pi}{3}} \), which complements the elegance complex numbers possess when handled in polar form.This approach can make division less cumbersome and helps correctly understand relations between two complex numbers.