Given $z_1 = 2(\cos 60^{\circ} + I \sin 60^{\circ}$\] And $z_2 = 8(\cos 150^{\circ} + I \sin 150^{\circ}$\], What Is $z_1 Z_2$?A. $3(\cos 210^{\circ} + I \sin 210^{\circ}$\]B. $3(\cos 90^{\circ} + I \sin

Introduction

Complex numbers are an extension of the real number system, which includes a new element, the imaginary unit, denoted by $i$. The imaginary unit is defined as the square root of $-1$, i.e., $i^2 = -1$. Complex numbers have numerous applications in mathematics, physics, engineering, and other fields. In this article, we will focus on the multiplication of complex numbers in polar form.

Polar Form of Complex Numbers

A complex number $z$ can be represented in polar form as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude (or modulus) of the complex number, and $\theta$ is the argument (or angle) of the complex number. The magnitude $r$ is given by $r = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number, respectively. The argument $\theta$ is given by $\theta = \tan^{-1} \left( \frac{b}{a} \right)$.

Multiplication of Complex Numbers in Polar Form

The multiplication of two complex numbers in polar form is given by the formula:

$z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$

where $z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$.

Example: Multiplication of Two Complex Numbers

Let's consider the two complex numbers:

$z_1 = 2(\cos 60^{\circ} + i \sin 60^{\circ})$

and

$z_2 = 8(\cos 150^{\circ} + i \sin 150^{\circ})$

We need to find the product $z_1 z_2$.

Step 1: Find the Magnitudes

The magnitude of $z_1$ is $r_1 = 2$, and the magnitude of $z_2$ is $r_2 = 8$.

Step 2: Find the Arguments

The argument of $z_1$ is $\theta_1 = 60^{\circ}$, and the argument of $z_2$ is $\theta_2 = 150^{\circ}$.

Step 3: Find the Sum of the Arguments

The sum of the arguments is $\theta_1 + \theta_2 = 60^{\circ} + 150^{\circ} = 210^{\circ}$.

Step 4: Find the Product

The product $z_1 z_2$ is given by:

$z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$

$= 2 \cdot 8 (\cos 210^{\circ} + i \sin 210^{\circ})$

$= 16 (\cos 210^{\circ} + i \sin 210^{\circ})$

Conclusion

In this article, we have discussed the multiplication of complex numbers in polar form. We have used the formula $z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$ to find the product of two complex numbers. We have also provided an example of the multiplication of two complex numbers, $z_1 = 2(\cos 60^{\circ} + i \sin 60^{\circ})$ and $z_2 = 8(\cos 150^{\circ} + i \sin 150^{\circ})$. The product $z_1 z_2$ is $16 (\cos 210^{\circ} + i \sin 210^{\circ})$.

Final Answer

Q&A: Complex Numbers Multiplication

Q: What is the formula for multiplying two complex numbers in polar form? A: The formula for multiplying two complex numbers in polar form is:

$z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$

Q: How do I find the magnitude of a complex number in polar form? A: To find the magnitude of a complex number in polar form, you need to find the square root of the sum of the squares of the real and imaginary parts. The magnitude is given by:

$r = \sqrt{a^2 + b^2}$

Q: How do I find the argument of a complex number in polar form? A: To find the argument of a complex number in polar form, you need to find the inverse tangent of the ratio of the imaginary part to the real part. The argument is given by:

$\theta = \tan^{-1} \left( \frac{b}{a} \right)$

Q: What is the difference between the magnitude and the argument of a complex number? A: The magnitude of a complex number is a measure of its size or distance from the origin, while the argument is a measure of its direction or angle from the positive real axis.

Q: Can I multiply complex numbers in rectangular form? A: Yes, you can multiply complex numbers in rectangular form. To do this, you need to multiply the real and imaginary parts separately and then combine the results.

Q: How do I multiply complex numbers in rectangular form? A: To multiply complex numbers in rectangular form, you need to follow the distributive property and multiply the real and imaginary parts separately. For example:

$(a + bi) (c + di) = (ac - bd) + (ad + bc)i$

Q: What is the relationship between complex numbers and trigonometry? A: Complex numbers and trigonometry are closely related. The trigonometric functions sine and cosine are used to represent the real and imaginary parts of complex numbers in polar form.

Q: Can I use complex numbers to solve real-world problems? A: Yes, complex numbers have numerous applications in real-world problems, such as electrical engineering, signal processing, and control systems.

Q: What are some common applications of complex numbers? A: Some common applications of complex numbers include:

• Electrical engineering: Complex numbers are used to analyze and design electrical circuits.
• Signal processing: Complex numbers are used to filter and analyze signals.
• Control systems: Complex numbers are used to design and analyze control systems.
• Quantum mechanics: Complex numbers are used to describe the behavior of particles in quantum systems.

Conclusion

In this article, we have discussed the multiplication of complex numbers in polar form and provided a step-by-step guide to solving problems involving complex numbers. We have also answered some common questions about complex numbers and their applications. We hope that this article has been helpful in understanding the concept of complex numbers and their importance in mathematics and real-world applications.