Problem 13 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=10\left(\cos 200^{\circ}+i \sin 200^{\circ}\right) \text { and } z_{2}=5\left(\cos 65^{\circ}+i \sin 65^{\circ}\right)$$

Step-by-Step Solution

Answer

The quotient is \(-\sqrt{2} + i\sqrt{2}\) in rectangular form.

Step 1: Convert to Exponential Form

Given the polar forms of the complex numbers, we express them in exponential form using Euler's formula: \[ z = r (\cos \theta + i \sin \theta) = r e^{i \theta} \]Thus, \[ z_{1} = 10e^{i 200^{\circ}} \] and \[ z_{2} = 5e^{i 65^{\circ}} \] .

Step 2: Calculate the Quotient

Divide the two exponential forms: \[ \frac{z_{1}}{z_{2}} = \frac{10e^{i 200^{\circ}}}{5e^{i 65^{\circ}}} \]Simplifying, we get: \[ \frac{10}{5} \times e^{i (200^{\circ} - 65^{\circ})} = 2 e^{i 135^{\circ}} \] .

Step 3: Convert to Rectangular Form

Now, convert \( 2 e^{i 135^{\circ}} \) back to rectangular form using Euler's formula:\[ 2 (\cos 135^{\circ} + i \sin 135^{\circ}) \]Substituting for the trigonometric values, we have:\[ 2 \left(-\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = -\sqrt{2} + i\sqrt{2} \] .

Key Concepts

Polar Form

Complex numbers can be expressed in multiple formats, and one of the most intuitive is the polar form. This form represents a complex number as a combination of a magnitude and an angle.

The polar form is usually written as \[ z = r \left( \cos \theta + i \sin \theta \right) \]where:

• \( r \) is the magnitude of the complex number and is calculated as the distance from the origin to the point on the complex plane.
• \( \theta \) is the angle (or argument) measured counterclockwise from the positive real axis to the line connecting the origin to the complex number.

This form is particularly useful when multiplying or dividing complex numbers, as it simplifies the arithmetic by separating the magnitude and angle.

The conversion to polar form involves trigonometric functions, which allows for a more graphic interpretation of complex numbers on the Argand diagram.

Exponential Form

The exponential form of a complex number leverages Euler's formula, which beautifully links together exponential functions and trigonometry: \[ e^{i \theta} = \cos \theta + i \sin \theta \] So, a complex number in polar form can be succinctly written as\[ z = r e^{i \theta} \] This notation is not just mathematically elegant, but it also simplifies many operations.

For example, multiplying two complex numbers in exponential form involves multiplying their magnitudes and adding their angles:

• \( z_{1} = r_{1} e^{i \theta_{1}} \)
• \( z_{2} = r_{2} e^{i \theta_{2}} \)
• \( z_{1} \cdot z_{2} = r_{1} r_{2} e^{i (\theta_{1} + \theta_{2})} \)

This forms an efficient way to handle power and root calculations, especially in contexts like signal processing or complex multiplication.

In the task you're working on, you converted complex numbers to exponential form to facilitate the division, as division involves subtracting the angles rather than manipulating trigonometric expressions directly.

Rectangular Form

In the rectangular form of complex numbers, the focus is on Cartesian coordinates such that a complex number is \[ z = a + bi \], where:

• \( a \) is the real part.
• \( b \) is the imaginary part, denoted by \( i \).

This form is particularly useful for straightforward algebraic operations such as addition and subtraction. Each part of the number directly corresponds to movement along the respective axis on the complex plane.

Converting from exponential or polar to rectangular form involves using trigonometric identities:

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

This conversion allows you to express results in terms that are more suitable for plotting or further algebraic manipulation, as seen when solving the initial exercise. By translating the result into rectangular form, it becomes easier to visualize and use in real-world applications. This method ensures accuracy and coherence in complex number calculations.