Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=-5+5 i, \quad z_{2}=-3 i $$

Step-by-Step Solution

Answer

\( z_1 z_2 = 15\sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \); \( \frac{z_1}{z_2} = \frac{5\sqrt{2}}{3} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \).

Step 1: Convert Complex Numbers to Polar Form

First, convert the given complex numbers to their polar forms. For \( z_1 = -5 + 5i \): - Calculate the modulus \( r_1 = \sqrt{(-5)^2 + 5^2} = \sqrt{50} = 5\sqrt{2} \).- Determine the argument \( \theta_1 = \tan^{-1}\left(\frac{5}{-5}\right) = \tan^{-1}(-1) = \frac{3\pi}{4} \) (since the complex number is in the second quadrant).- Thus, \( z_1 = 5\sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \).For \( z_2 = -3i \):- The modulus \( r_2 = 3 \).- The argument \( \theta_2 = \tan^{-1}\left(\frac{-3}{0}\right) = -\frac{\pi}{2} \) (since it points downwards on the imaginary axis).- Thus, \( z_2 = 3 ( \cos(-\frac{\pi}{2}) + i \sin(-\frac{\pi}{2}) ) \).

Step 2: Multiply the Complex Numbers

Use the trigonometric form to multiply the complex numbers. The formula to multiply two complex numbers in polar form is:\[ z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right) \]Substitute the values:- \( r_1 = 5\sqrt{2} \), \( \theta_1 = \frac{3\pi}{4} \)- \( r_2 = 3 \), \( \theta_2 = -\frac{\pi}{2} \)Calculate the product:- Modulus: \( 15\sqrt{2} \)- Argument: \( \frac{3\pi}{4} - \frac{\pi}{2} = \frac{\pi}{4} \)\( z_1 z_2 = 15\sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \)

Step 3: Divide the Complex Numbers

Use the trigonometric form to divide the complex numbers. The formula to divide two complex numbers in polar form is:\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \]Substitute the values:- \( r_1 = 5\sqrt{2} \), \( \theta_1 = \frac{3\pi}{4} \)- \( r_2 = 3 \), \( \theta_2 = -\frac{\pi}{2} \)Calculate the division:- Modulus: \( \frac{5\sqrt{2}}{3} \)- Argument: \( \frac{3\pi}{4} + \frac{\pi}{2} = \frac{9\pi}{4} = \frac{\pi}{4} + 2\pi \)Simplifying, \( \frac{z_1}{z_2} = \frac{5\sqrt{2}}{3} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \).

Key Concepts

Trigonometric Form of Complex Numbers

Complex numbers can initially seem a bit abstract, but when we describe them using a trigonometric form, they become more manageable. This form relies on two main components: the modulus and the argument.
In trigonometric form, a complex number can be written as:

• \( z = r (\cos \theta + i \sin \theta) \)

Here, \( r \) is the modulus, representing the distance from the origin in the complex plane, while \( \theta \) is the argument, indicating the angle from the positive real axis.Converting a complex number to its trigonometric form is particularly useful for multiplication and division operations, making it easier to handle them systematically.

Polar Form Conversion

Turning a complex number from its standard form into polar form involves a few straightforward steps. Let's break it down.
For example, take a complex number \( z = a + bi \):

• First, find the modulus \( r = \sqrt{a^2 + b^2} \).
• Next, determine the argument \( \theta \) using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Ensure you identify the correct quadrant, as this affects the sign and value of \( \theta \).

Once you have \( r \) and \( \theta \), you can express the number in polar form as \( r(\cos \theta + i \sin \theta) \).
This conversion makes it clear that each complex number can be viewed as a combination of a rotation (via \( \theta \)) and a scaling (via \( r \)).

Modulus and Argument in Complex Numbers

In complex numbers, the modulus and argument are essential concepts that help us visualize and calculate with these numbers more easily.
The **modulus** of a complex number \( z = a + bi \) is its distance from the origin in the complex plane, calculated as:

• \( |z| = \sqrt{a^2 + b^2} \)

The **argument** of \( z \) is the angle \( \theta \) formed with the positive real axis, guiding us in understanding its direction:

• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

Remember that the argument requires careful attention to the quadrant where the number lies to ensure the angle is measured correctly. This understanding underpins more complex operations like multiplication and division in polar format.

Complex Number Operations

Operations on complex numbers can become straightforward by utilizing their polar forms. Let's see how multiplication and division function here.

Multiplication

When you multiply two complex numbers in polar form \( r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( r_2 (\cos \theta_2 + i \sin \theta_2) \):

• Multiply the moduli: \( r = r_1 \times r_2 \)
• Add the arguments: \( \theta = \theta_1 + \theta_2 \)

The result is \( r (\cos \theta + i \sin \theta) \).

Division

For dividing \( z_1 \) by \( z_2 \) using their polar forms:

• Divide the moduli: \( r = \frac{r_1}{r_2} \)
• Subtract the arguments: \( \theta = \theta_1 - \theta_2 \)

Thus, the quotient becomes \( r (\cos \theta + i \sin \theta) \).
Using these forms simplifies complex number operations by reducing them to arithmetic operations on the moduli and trigonometric operations on the arguments.