Problem 62 - Algebra and Trigonometry

Question

\(57-64=\) Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=4 \sqrt{3}-4 i, \quad z_{2}=8 i $$

Step-by-Step Solution

Answer

Product: 32 + 32i√3, Quotient: -1/2 - i√3/2, Reciprocal: √3/16 + i/16.

Step 1: Convert to polar form

First, we find the polar form of \( z_1 = 4\sqrt{3} - 4i \). Calculate its magnitude: \( r_1 = \sqrt{(4\sqrt{3})^2 + (-4)^2} = \sqrt{48 + 16} = \sqrt{64} = 8 \). The argument \( \theta_1 \) is given by \( \tan^{-1} \left(\frac{-4}{4\sqrt{3}}\right) = \tan^{-1} \left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \). So, \( z_1 = 8(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6})) \).

Step 2: Convert second complex number to polar

For \( z_2 = 8i \), its magnitude is \( r_2 = \sqrt{0^2 + 8^2} = \sqrt{64} = 8 \), and its argument is \( \theta_2 = \frac{\pi}{2} \) since it lies on the positive imaginary axis. So, \( z_2 = 8(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})) \).

Step 3: Find the product \( z_1 z_2 \)

The product of two complex numbers in polar form is found by multiplying their magnitudes and adding their arguments. Therefore, \( z_1z_2 = (8 \times 8)[\cos(-\frac{\pi}{6} + \frac{\pi}{2}) + i\sin(-\frac{\pi}{6} + \frac{\pi}{2})] = 64[\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})] = 64[\frac{1}{2} + i\frac{\sqrt{3}}{2}] = 32 + 32i\sqrt{3} \).

Step 4: Find the quotient \( \frac{z_1}{z_2} \)

To find the quotient, divide the magnitudes and subtract the arguments: \( \frac{z_1}{z_2} = \frac{8}{8}[\cos(-\frac{\pi}{6} - \frac{\pi}{2}) + i\sin(-\frac{\pi}{6} - \frac{\pi}{2})] = 1[\cos(-\frac{2\pi}{3}) + i\sin(-\frac{2\pi}{3})] = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \).

Step 5: Find \( \frac{1}{z_1} \)

To find \( \frac{1}{z_1} \), take the reciprocal of the magnitude and change the sign of the argument: \( \frac{1}{z_1} = \frac{1}{8}[\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6})] = \frac{1}{8}[\frac{\sqrt{3}}{2} + i\frac{1}{2}] = \frac{\sqrt{3}}{16} + \frac{i}{16} \).

Key Concepts

Polar Form

Complex numbers can be expressed in rectangular form (as used in math class), or in polar form, which is useful for multiplying and dividing them.
In polar form, a complex number is represented as its magnitude (distance from the origin) and angle (direction from the positive x-axis) in the complex plane.
A complex number in polar form is written as:

• \( z = r(\cos(\theta) + i\sin(\theta)) \)
• "\(r\)" is the magnitude.
• "\(\theta\)" is the argument (angle).

Using Euler’s formula, it can be simplified to: \( z = re^{i\theta} \).
It makes operations like multiplication and division simpler, as we only need to work with the magnitudes and angles.

Magnitude

The magnitude, or modulus, of a complex number tells us how far the point is from the origin in the complex plane.
For a complex number \( z = a + bi \), its magnitude is:
\[ r = \sqrt{a^2 + b^2} \]
If you imagine the complex plane as a graph, the magnitude is the length of the line from the origin (0,0) to the point \((a,b)\).
For the given exercise, we calculated the magnitudes as:

• \( r_1 = 8 \) for \( z_1 = 4\sqrt{3} - 4i \)
• \( r_2 = 8 \) for \( z_2 = 8i \)

This step is crucial before moving to polar form.

Argument

The argument of a complex number refers to the angle it makes with the positive x-axis in the complex plane.
This angle is usually measured in radians, ranging from \(0\) to \(2\pi\) for a full circle.
For a complex number \( z = a + bi \), the argument \( \theta \) is calculated using:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
Adjustments might be needed based on the quadrant in which the complex number lies.
In our example, the arguments calculated are:

• \( \theta_1 = -\frac{\pi}{6} \) for \( z_1 = 4\sqrt{3} - 4i \)
• \( \theta_2 = \frac{\pi}{2} \) for \( z_2 = 8i \)

Understanding the argument helps pinpoint the direction of the number.

Complex Multiplication

Complex multiplication is simplified by expressing numbers in polar form.
When multiplying two complex numbers in polar form, you:

• Multiply their magnitudes: \( r_1r_2 \)
• Add their arguments: \( \theta_1 + \theta_2 \)

This is due to the properties of sine and cosine during multiplication, making it more intuitive and streamlined.
In the example, for multiplying \( z_1 \) and \( z_2 \), the product is expressed as:\[ z_1z_2 = 64 \left( \cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}) \right) \]
Resulting in the rectangular form of \( 32 + 32i\sqrt{3} \). Understanding this helps apply the concept of using polar coordinates efficiently in complex operations.