Problem 14 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=4\left(\cos 280^{\circ}+i \sin 280^{\circ}\right) \text { and } z_{2}=4\left(\cos 55^{\circ}+i \sin 55^{\circ}\right)$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}\).

Step 1: Identify Given Complex Numbers

We have two complex numbers in trigonometric form: 1. \( z_1 = 4(\cos 280^{\circ} + i \sin 280^{\circ}) \)2. \( z_2 = 4(\cos 55^{\circ} + i \sin 55^{\circ}) \)

Step 2: Apply the Division Formula for Complex Numbers in Polar Form

The quotient \( \frac{z_1}{z_2} \) in polar form is given by:\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \]For this problem:\[ \frac{z_1}{z_2} = \frac{4}{4} \left( \cos(280^{\circ} - 55^{\circ}) + i \sin(280^{\circ} - 55^{\circ}) \right) \]

Step 3: Simplify the Expression

Since \( r_1 = r_2 = 4 \), the magnitudes cancel out, leaving:\[ z = \cos(225^{\circ}) + i \sin(225^{\circ}) \]

Step 4: Convert from Polar to Rectangular Form

To express the result in rectangular form, convert \( \cos(225^{\circ}) \) and \( \sin(225^{\circ}) \) using known trigonometric values:\[ \cos(225^{\circ}) = -\frac{\sqrt{2}}{2}, \quad \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} \]Thus, the rectangular form is:\[ z = -\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \]

Step 5: Final Answer

The quotient \( \frac{z_1}{z_2} \) in rectangular form is:\[ -\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \]

Key Concepts

Rectangular Form

Complex numbers can be expressed in several forms, one of which is the rectangular form. This is the standard way we usually see complex numbers written, as they consist of a real part and an imaginary part. For example, consider a complex number written as \( a + bi \), where:

• \( a \) is the real part.
• \( bi \) is the imaginary part.

In the exercise, after dividing the complex numbers \( z_1 \) and \( z_2 \), the quotient is converted to rectangular form. This involves evaluating \( \cos(225^{\circ}) \) and \( \sin(225^{\circ}) \) to find the real and imaginary components. By substituting the values \( \cos(225^{\circ}) = -\frac{\sqrt{2}}{2} \) and \( \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} \), we obtain the rectangular result: \(-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).
This form is particularly useful because it directly aligns with how we often interpret points or vectors in a two-dimensional plane. It shows both horizontal and vertical changes clearly. Understanding and being able to convert complex numbers into their rectangular form is crucial because it makes them easier to visualize and manipulate, especially when adding or subtracting complex numbers.

Polar Form

The polar form of a complex number is used extensively in mathematics because of its ability to simplify multiplication, division, and the taking of powers and roots of complex numbers. In this form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \), where:

• \( r \) is the magnitude (or absolute value) of the complex number.
• \( \theta \) is the angle, known as the argument, measured from the positive x-axis to the line segment representing the complex number.

In the original exercise, both \( z_1 \) and \( z_2 \) are initially given in polar form. This form is directly tied to the unit circle and is very intuitive when dealing with multiplicative operations.
When dividing complex numbers in polar form, as done in the solution, the magnitudes are divided, and the angles are subtracted: \[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \]. Here, since the magnitudes \( r_1 \) and \( r_2 \) are equal, they cancel out, and we mainly focus on the result of the angle subtraction. This simplification is one of the primary advantages of using polar form.

Trigonometric Form

Trigonometric form, often synonymous with polar form, is a way of expressing complex numbers to emphasize their real and imaginary components through trigonometric functions. In this form, a complex number \( z \) is expressed as: \[ z = r (\cos \theta + i \sin \theta) \] where:

• \( r \) indicates the modulus, which is the distance from the origin to the point \( z \) in the complex plane.
• \( \theta \) represents the angle (or argument) formed with the positive x-axis.

Understanding trigonometric form is key to comprehending how complex numbers behave geometrically. It simplifies operations such as multiplication and division in the complex plane by transforming them into operations on angles and magnitudes.
For instance, in the division of two numbers \( z_1 \) and \( z_2 \) given in trigonometric form, the concept of subtracting angles (\( \theta_1 - \theta_2 \)) simplifies the calculation, as seen in the exercise. Furthermore, understanding these fundamentals supports converting them back into forms that fit contexts where combinations are easier or more consistent to work with, like the rectangular form for addition/subtraction.