Problem 49 - Algebra and Trigonometry

Question

\(49-56\) me 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: Interpret the given complex numbers

The complex numbers are given in trigonometric form where \( z_1 = \cos \pi + i \sin \pi \). This corresponds to \((\cos\theta + i\sin\theta)\) which is equivalent to the polar form \( e^{i\theta} \). Hence, \( z_1 = e^{i\pi} \).For \( z_2 = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \), using the same reasoning, \( z_2 = e^{i\frac{\pi}{3}} \).

Step 2: Find the product \(z_1 z_2\)

The product of two complex numbers in polar form is simply the product of their magnitudes and the sum of their angles. Thus, \( z_1 z_2 = e^{i\pi} \times e^{i\frac{\pi}{3}} = e^{i(\pi + \frac{\pi}{3})} = e^{i\frac{4\pi}{3}} \).

Step 3: Find the quotient \(z_1 / z_2\)

The quotient of two complex numbers in polar form is the quotient of their magnitudes and the difference of their angles. Thus, \( \frac{z_1}{z_2} = \frac{e^{i\pi}}{e^{i\frac{\pi}{3}}} = e^{i(\pi - \frac{\pi}{3})} = e^{i\frac{2\pi}{3}} \).

Step 4: Express product and quotient in polar form

The product \( z_1 z_2 = e^{i\frac{4\pi}{3}} \) is in polar form with angle \( \frac{4\pi}{3} \).The quotient \( \frac{z_1}{z_2} = e^{i\frac{2\pi}{3}} \) is in polar form with angle \( \frac{2\pi}{3} \).

Key Concepts

Polar Form

Polar form is a way to represent complex numbers using a magnitude (or modulus) and an angle. Instead of conveying complex numbers as a sum of real and imaginary parts, polar form provides a geometric perspective. You can picture complex numbers as points in the complex plane, with the x-axis representing real numbers and the y-axis for imaginary numbers. The polar form is written as \[ z = re^{i\theta} \]where:

• \( r \) is the modulus of the complex number, representing the distance from the origin to the point.
• \( \theta \) is the argument of the complex number, the angle formed with the positive x-axis.

The beauty of polar form is its simplicity when dealing with multiplication and division of complex numbers, as we'll see in the following sections. For expressiveness, polar form effectively combines the magnitude and direction of a vector in the complex plane.

Trigonometric Form

The trigonometric form of a complex number is closely related to polar form. It's represented using a combination of cosine and sine functions:\[ z = r(\cos\theta + i\sin\theta) \]This notation emphasizes the coordinates on the unit circle:

• The cosine function \( \cos\theta \) gives the x-coordinate, or the real part.
• The sine function \( \sin\theta \) gives the y-coordinate, or the imaginary part.

In practice, trigonometric form is mainly useful for converting complex numbers to polar form, as it shows explicitly how \( r \) and \( \theta \) relate to the familiar sine and cosine from trigonometric studies. By understanding b th representations, you gain flexibility to switch between different forms conveniently when solving problems.

Complex Number Multiplication

Complex number multiplication is greatly simplified in polar form. When you multiply two complex numbers written in polar form, you simply multiply their magnitudes and add their angles:\[ z_1 \times z_2 = r_1r_2e^{i(\theta_1 + \theta_2)} \]For example, in our exercise, the multiplication \( z_1 \times z_2 \) leads to:

• The sum of the angles \( \theta_1 = \pi \) and \( \theta_2 = \frac{\pi}{3} \), thus giving an angle of \( \frac{4\pi}{3} \).

This method bypasses the lengthier straightforward multiplication of binomial expressions that would be necessary in rectangular form. By using polar form, we simplify calculations and focus on understanding the geometric transformations represented by multiplication.

Complex Number Division

When dividing complex numbers in polar form, the operation is intuitive and concise. You divide the magnitudes and subtract the angles:\[ \frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)} \]From the exercise, for your divisors \( z_1 \) and \( z_2 \), division results in:

• The difference between the angles \( \theta_1 = \pi \) and \( \theta_2 = \frac{\pi}{3} \), which results in an angle of \( \frac{2\pi}{3} \).

This subtraction of angles reflects the rotational difference in the complex plane, providing both a simple calculation method and a geometric interpretation. Polar form thus makes both multiplication and division operations more straightforward, emphasizing the rotational dynamics in the complex plane.