Problem 11 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=6\left(\cos 100^{\circ}+i \sin 100^{\circ}\right) \text { and } z_{2}=2\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)$$

Step-by-Step Solution

Answer

\(\frac{z_1}{z_2} = \frac{3}{2} + i\frac{3\sqrt{3}}{2}\)

Step 1: Express in Polar Form

The given complex numbers \(z_1 = 6(\cos 100^{\circ} + i\sin 100^{\circ})\) and \(z_2 = 2(\cos 40^{\circ} + i\sin 40^{\circ})\) are already in polar form. In this form, \(z = r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude, and \(\theta\) is the angle.

Step 2: Compute the Quotient in Polar Form

To find \(\frac{z_1}{z_2}\), use the division property for complex numbers in polar form: \[\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)).\] Here, \(r_1 = 6\), \(r_2 = 2\), and the angles are \(\theta_1 = 100^{\circ}\) and \(\theta_2 = 40^{\circ}\).

Step 3: Calculate the Magnitude Ratio

The magnitude of the quotient is given by \(\frac{r_1}{r_2} = \frac{6}{2} = 3\).

Step 4: Calculate the Angle Difference

The angle for the quotient is \(\theta_1 - \theta_2 = 100^{\circ} - 40^{\circ} = 60^{\circ}\).

Step 5: Express the Quotient in Polar Form

Using the results from Steps 3 and 4, express \(\frac{z_1}{z_2}\) in polar form as \[3(\cos 60^{\circ} + i \sin 60^{\circ}).\]

Step 6: Convert Polar to Rectangular Form

To express the quotient in rectangular form, calculate \(\cos 60^{\circ} = \frac{1}{2}\) and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). So, the rectangular form is: \[3\left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = \frac{3}{2} + i\frac{3\sqrt{3}}{2}.\]

Key Concepts

Polar Coordinates

The concept of polar coordinates is crucial when dealing with complex numbers, especially in the field of mathematics and engineering. Polar coordinates represent points in the plane using a radius and an angle. This way of expressing complex numbers is different from the more typical rectangular form, which uses real and imaginary parts. Polar coordinates describe a complex number as \( z = r(\cos \theta + i \sin \theta) \), where:

• \( r \) is the magnitude (or modulus) of the number, essentially how far the number is from the origin (0,0).
• \( \theta \) is the angle (or argument) formed with the positive x-axis, showing the direction of the vector.

In our problem, both complex numbers are given in polar form: \( z_1 = 6(\cos 100^{\circ} + i\sin 100^{\circ}) \) and \( z_2 = 2(\cos 40^{\circ} + i\sin 40^{\circ}) \). To find their quotient, this representation is very useful as it allows us to use properties of exponentials and trigonometric functions directly.

Rectangular Form

The rectangular form of a complex number is perhaps the most familiar form for many, especially those starting out with complex arithmetic. It expresses complex numbers as \( z = a + bi \), where:

• \( a \) is the real part, corresponding to the x-coordinate.
• \( b \) is the imaginary part, corresponding to the y-coordinate.

After finding the quotient \( \frac{z_1}{z_2} \) in polar form, we converted it to rectangular form. This involved calculating the cosine and sine of the angle \( 60^{\circ} \). The results were \( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). Substituting these into our polar-form quotient, we arrived at the rectangular form: \( \frac{3}{2} + i\frac{3\sqrt{3}}{2} \). This step-by-step transition helps visualize complex numbers in relation to the coordinate plane.

Magnitude and Angle

Understanding the magnitude and angle is essential when working with complex numbers in polar form. The magnitude \( r \) represents the distance from the origin to the point in the complex plane, dictating the size of the number. The angle \( \theta \) gives the direction of the vector.
In the problem, calculating the magnitude of the quotient \( \frac{z_1}{z_2} \) was straightforward, as we simply divided the magnitudes of \( z_1 \) and \( z_2 \), resulting in a magnitude of 3. Similarly, computing the angle involved subtracting \( \theta_2 \) from \( \theta_1 \), giving us \( 60^{\circ} \). This angle difference allowed us to further convert the result into rectangular form, completing our task. Understanding these concepts makes manipulating complex numbers significantly easier, particularly when solving problems involving division or multiplication.