Problem 10 - Precalculus

Question

Find the product \(z_{1} z_{2}\) and express it in rectangular form. $$z_{1}=6\left[\cos \left(\frac{2 \pi}{9}\right)+i \sin \left(\frac{2 \pi}{9}\right) \text { and } z_{2}=5\left[\cos \left(\frac{\pi}{9}\right)+i \sin \left(\frac{\pi}{9}\right)\right]\right.$$

Step-by-Step Solution

Answer

The product in rectangular form is \( 15 + 15\sqrt{3}i \).

Step 1: Understand the Given Complex Numbers

The complex numbers are given in polar form. Specifically, \( z_1 = 6\left[\cos \left(\frac{2\pi}{9}\right) + i \sin \left(\frac{2\pi}{9}\right)\right] \) and \( z_2 = 5\left[\cos \left(\frac{\pi}{9}\right) + i \sin \left(\frac{\pi}{9}\right)\right] \). The polar form expression is \( r[\cos\theta + i\sin\theta] \), better known as \( r e^{i\theta} \).

Step 2: Multiply the Complex Numbers

To multiply complex numbers in polar form, \( z_1 z_2 = r_1 e^{i\theta_1} \times r_2 e^{i\theta_2} = (r_1r_2)e^{i(\theta_1 + \theta_2)} \). Here, \( r_1 = 6 \), \( \theta_1 = \frac{2\pi}{9} \); \( r_2 = 5 \), \( \theta_2 = \frac{\pi}{9} \). Multiply magnitudes: \( 6 \times 5 = 30 \), and add angles: \( \frac{2\pi}{9} + \frac{\pi}{9} = \frac{3\pi}{9} = \frac{\pi}{3} \). So, \( z_1 z_2 = 30[\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})] \).

Step 3: Convert to Rectangular Form

The rectangular form of a complex number is \( a + bi \), where \( a = r \cos \theta \) and \( b = r \sin \theta \). For \( \theta = \frac{\pi}{3} \), \( \cos(\frac{\pi}{3}) = \frac{1}{2} \) and \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \). Compute \( a = 30 \times \frac{1}{2} = 15 \) and \( b = 30 \times \frac{\sqrt{3}}{2} = 15\sqrt{3} \). Thus, the rectangular form is \( 15 + 15\sqrt{3}i \).

Key Concepts

Polar Form of Complex Numbers

When it comes to complex numbers, the polar form is a powerful way to represent them. A complex number in polar form is expressed as

• \( z = r \left[\cos \theta + i\sin \theta\right] \)
• or as its equivalent, using Euler's formula: \( z = re^{i\theta} \).

This representation is particularly useful for multiplication and division tasks within the realm of complex numbers. Here's why: In polar form,

• \( r \) represents the magnitude or the distance from the origin to the point on the complex plane.
• \( \theta \) stands for the angle, often called the argument, that the line connecting the origin to the point makes with the positive x-axis.

Using the polar form allows us to multiply two complex numbers easily by multiplying their magnitudes (\( r_1 \times r_2 \)) and adding their angles (\( \theta_1 + \theta_2 \)). This makes the complex plane transformations intuitive and straightforward. In the problem, the numbers are given as \( z_1 = 6\left[\cos \left(\frac{2\pi}{9}\right) + i \sin \left(\frac{2\pi}{9}\right)\right] \) and \( z_2 = 5\left[\cos \left(\frac{\pi}{9}\right) + i \sin \left(\frac{\pi}{9}\right)\right] \). Their product is easily computed in this form.

Rectangular Form of Complex Numbers

The rectangular form, also known as the Cartesian form, expresses a complex number as \( z = a + bi \). Here \( a \) and \( b \) are real numbers, where

• \( a \) is called the real part,
• \( b \) is called the imaginary part.

Converting complex numbers from polar to rectangular form requires using the angle \( \theta \) to determine the components \( a \) and \( b \). Using the relationships:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

For example, when we multiply two polar complex numbers as in the given exercise, the resulting number in polar form is \( 30[\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})] \). To convert this back into rectangular form, we calculate:

• \( a = 30 \times \frac{1}{2} = 15 \)
• \( b = 30 \times \frac{\sqrt{3}}{2} = 15\sqrt{3} \)

Thus, the rectangular form of the product is \( 15 + 15\sqrt{3}i \), which combines both the real and imaginary components.

Trigonometric Identities

Trigonometric identities are vital mathematical tools that play a crucial role in computing with complex numbers, especially in conversions and simplifications.

• They help us relate different trigonometric functions with each other.
• They are extensively used in expressing complex numbers given in trigonometric form.

For example, the angle addition formulas,

• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)

are particularly useful in complex multiplication when handling products like \( z_1 z_2 = (r_1r_2) (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \). In our example, to find the resulting angle \( \frac{\pi}{3} \), we simply added \( \frac{2\pi}{9} \) and \( \frac{\pi}{9} \), demonstrating the simple computation enabled by these identities. Understanding and applying these identities effectively can greatly simplify many problems in trigonometry and complex analysis.