Finding Real And Imaginary Parts Of Complex Number Division If $z_1 = 2 Cis 120^{\circ}$, $z_2 = 4 Cis 30^{\circ}$, And $\frac{z_1}{z_2} = A + Bi$

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Introduction to Complex Numbers in Polar Form

In the realm of mathematics, complex numbers extend the real number system by including an imaginary unit, denoted as i, which is defined as the square root of -1. A complex number is generally expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The polar form, also known as the trigonometric form, provides an alternative representation of complex numbers that is particularly useful for understanding their geometric properties and performing operations such as multiplication and division.

The polar form of a complex number z is given by z = r cis θ, where r is the magnitude (or modulus) of z, θ is the argument of z, and cis θ is a shorthand notation for cos θ + i sin θ. The magnitude r represents the distance from the origin to the point representing z in the complex plane, while the argument θ represents the angle between the positive real axis and the line connecting the origin to the point representing z. The complex plane is a geometric representation of complex numbers, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

Understanding the polar form of complex numbers is crucial for simplifying operations like multiplication and division. When multiplying complex numbers in polar form, you multiply their magnitudes and add their arguments. Conversely, when dividing complex numbers in polar form, you divide their magnitudes and subtract their arguments. These rules make complex number arithmetic more intuitive and efficient, especially in fields like electrical engineering and quantum mechanics, where complex numbers are frequently used to model oscillatory phenomena and wave behavior.

Problem Statement and Initial Setup

In this article, we delve into a problem involving the division of complex numbers expressed in polar form. Given two complex numbers, $z_1 = 2 \operatorname{cis} 120^{\circ}$ and $z_2 = 4 \operatorname{cis} 30^{\circ}$, our task is to find the quotient $\frac{z_1}{z_2}$ and express it in the standard rectangular form a + bi, where a and b are real numbers. We are asked to determine the values of a and b.

Here, the notation $\operatorname{cis} θ$ is a shorthand for cos θ + i sin θ. Thus, we have:

$z_1 = 2(\cos 120^{\circ} + i \sin 120^{\circ})$

$z_2 = 4(\cos 30^{\circ} + i \sin 30^{\circ})$

To solve this problem, we will use the property of complex numbers in polar form that states when dividing two complex numbers, their magnitudes are divided, and their arguments are subtracted. This property stems directly from the trigonometric identities and the properties of complex exponentials, making the division operation straightforward when the numbers are expressed in polar form.

Step-by-Step Solution for Dividing Complex Numbers

To find $\frac{z_1}{z_2}$, we will use the division rule for complex numbers in polar form. This involves dividing the magnitudes and subtracting the arguments:

$\frac{z_1}{z_2} = \frac{2 \operatorname{cis} 120^{\circ}}{4 \operatorname{cis} 30^{\circ}}$

First, we divide the magnitudes:

$\frac{|z_1|}{|z_2|} = \frac{2}{4} = \frac{1}{2}$

Next, we subtract the arguments:

$\arg(\frac{z_1}{z_2}) = 120^{\circ} - 30^{\circ} = 90^{\circ}$

So, $\frac{z_1}{z_2} = \frac{1}{2} \operatorname{cis} 90^{\circ}$.

Now, we convert this back to the rectangular form a + bi:

$\frac{1}{2} \operatorname{cis} 90^{\circ} = \frac{1}{2}(\cos 90^{\circ} + i \sin 90^{\circ})$

Since $\cos 90^{\circ} = 0$ and $\sin 90^{\circ} = 1$, we have:

$\frac{1}{2}(0 + i(1)) = \frac{1}{2}i$

Therefore, $\frac{z_1}{z_2} = 0 + \frac{1}{2}i$.

Determining the Values of a and b

From the previous step, we have found that $\frac{z_1}{z_2} = 0 + \frac{1}{2}i$. Comparing this to the form a + bi, we can directly identify the values of a and b.

The real part, a, is the coefficient of the real term, which in this case is 0.

The imaginary part, b, is the coefficient of the imaginary unit i, which in this case is $\frac{1}{2}$.

Thus, we have:

a = 0

b = $\frac{1}{2}$

These values represent the real and imaginary components of the quotient $\frac{z_1}{z_2}$ when expressed in the standard rectangular form. Understanding how to manipulate complex numbers in polar form allows for efficient calculation and simplification of complex expressions, which is a fundamental skill in various mathematical and scientific disciplines.

Verification and Alternative Methods

To verify our solution, we can convert the complex numbers back to rectangular form individually and then perform the division. This will give us an alternative method to check our result. Initially, we have:

$z_1 = 2(\cos 120^{\circ} + i \sin 120^{\circ})$

$z_2 = 4(\cos 30^{\circ} + i \sin 30^{\circ})$

We know that $\cos 120^{\circ} = -\frac{1}{2}$, $\sin 120^{\circ} = \frac{\sqrt{3}}{2}$, $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$, and $\sin 30^{\circ} = \frac{1}{2}$. Substituting these values, we get:

$z_1 = 2(-\frac{1}{2} + i \frac{\sqrt{3}}{2}) = -1 + i\sqrt{3}$

$z_2 = 4(\frac{\sqrt{3}}{2} + i \frac{1}{2}) = 2\sqrt{3} + 2i$

Now, we divide $z_1$ by $z_2$:

$\frac{z_1}{z_2} = \frac{-1 + i\sqrt{3}}{2\sqrt{3} + 2i}$

To divide, we multiply the numerator and denominator by the conjugate of the denominator:

$\frac{-1 + i\sqrt{3}}{2\sqrt{3} + 2i} \cdot \frac{2\sqrt{3} - 2i}{2\sqrt{3} - 2i}$

Expanding the numerator:

$(-1 + i\sqrt{3})(2\sqrt{3} - 2i) = -2\sqrt{3} + 2i + 6i - 2i^2\sqrt{3} = -2\sqrt{3} + 8i + 2\sqrt{3} = 8i$

Expanding the denominator:

$(2\sqrt{3} + 2i)(2\sqrt{3} - 2i) = 12 - 4i^2 = 12 + 4 = 16$

So, $\frac{z_1}{z_2} = \frac{8i}{16} = \frac{1}{2}i$, which matches our previous result of $0 + \frac{1}{2}i$. This confirms that our solution is correct.

Conclusion and Key Takeaways

In conclusion, given $z_1 = 2 \operatorname{cis} 120^{\circ}$ and $z_2 = 4 \operatorname{cis} 30^{\circ}$, we found that $\frac{z_1}{z_2} = 0 + \frac{1}{2}i$. Therefore, the values are:

a = 0

b = $\frac{1}{2}$

This problem demonstrates the utility of the polar form in simplifying complex number operations, particularly division. By converting complex numbers to polar form, dividing them becomes a straightforward process of dividing magnitudes and subtracting arguments. The result can then be easily converted back to the rectangular form a + bi if needed.

Understanding complex number division is essential in various fields, including electrical engineering, where complex numbers are used to analyze AC circuits, and quantum mechanics, where they describe wave functions. The ability to manipulate complex numbers efficiently allows for simpler solutions to problems in these domains. This article provided a comprehensive step-by-step solution, along with a verification method, to ensure a clear understanding of the process. By mastering these techniques, students and professionals can confidently tackle more complex problems involving complex numbers.