Problem 3 - Precalculus

Question

Find the product \(z_{1} z_{2}\) and express it in rectangular form. $$z_{1}=4\left(\cos 80^{\circ}+i \sin 80^{\circ}\right) \text { and } z_{2}=2\left(\cos 145^{\circ}+i \sin 145^{\circ}\right)$$

Step-by-Step Solution

Answer

The product is \(-4\sqrt{2} - 4i\sqrt{2}\) in rectangular form.

Step 1: Recognize the Polar Form

Identify that both complex numbers are given in polar form: \( z_{1} = 4(\cos 80^{\circ} + i \sin 80^{\circ}) \) and \( z_{2} = 2(\cos 145^{\circ} + i \sin 145^{\circ}) \). In this form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

Step 2: Multiply Moduli of the Complex Numbers

The modulus of the product of two complex numbers is the product of their moduli. Calculate the moduli: \( |z_{1}| = 4 \) and \( |z_{2}| = 2 \). The modulus of the product is \( |z_{1}z_{2}| = 4 \times 2 = 8 \).

Step 3: Add the Arguments

The argument of the product of two complex numbers is the sum of their arguments. Calculate: \( \text{arg}(z_{1}) = 80^{\circ} \) and \( \text{arg}(z_{2}) = 145^{\circ} \). Add these to get the argument of the product: \( \text{arg}(z_{1}z_{2}) = 80^{\circ} + 145^{\circ} = 225^{\circ} \).

Step 4: Convert Product to Rectangular Form

Now, express \( z_{1}z_{2} \) using its modulus and argument in rectangular form. We have: \( z_{1}z_{2} = 8(\cos 225^{\circ} + i \sin 225^{\circ}) \). Calculate \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \) and \( \sin 225^{\circ} = -\frac{\sqrt{2}}{2} \). Thus, \( z_{1}z_{2} = 8 \left(-\frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right)\right) \).

Step 5: Simplify the Expression

Simplify the expression: \( z_{1}z_{2} = 8 \left(-\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}\right) = 8 \cdot -\frac{\sqrt{2}}{2} - 8i \cdot \frac{\sqrt{2}}{2} = -4\sqrt{2} - 4i\sqrt{2} \). Thus, the rectangular form is \( -4\sqrt{2} - 4i\sqrt{2} \).

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be expressed in various forms, with polar form being one of the most versatile and insightful. In polar form, a complex number is represented as \( r (\cos \theta + i \sin \theta) \). Here, \( r \) stands for the modulus, while \( \theta \) is the argument or angle. This representation allows you to easily visualize complex numbers as points in the polar coordinate system on a plane.
In our exercise, the complex numbers \( z_1 \) and \( z_2 \) are originally given in polar form:

• \( z_1 = 4(\cos 80^{\circ} + i \sin 80^{\circ}) \)
• \( z_2 = 2(\cos 145^{\circ} + i \sin 145^{\circ}) \)

Polar form is particularly useful for multiplication and division of complex numbers. To multiply two complex numbers, you simply multiply their moduli \( r_1 \times r_2 \) and add their arguments \( \theta_1 + \theta_2 \). This makes complex arithmetic clean and straightforward in the polar domain.

Rectangular Form of Complex Numbers

The rectangular form of complex numbers is another common way to express them, written as \( a + bi \). Here, \( a \) and \( b \) are real numbers, representing the real and imaginary parts of the complex number respectively.
In many practical applications, converting a complex number from polar to rectangular form is necessary for computational purposes. This involves using the trigonometric identities for cosine and sine along with the modulus to find the values of \( a \) and \( b \).
In the original exercise, after finding the product of \( z_1 \) and \( z_2 \) in polar form \(8(\cos 225^{\circ} + i \sin 225^{\circ})\), it is converted into rectangular form by calculating \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \) and \( \sin 225^{\circ} = -\frac{\sqrt{2}}{2}\). Thus:

• \( a = 8 \times \left( -\frac{\sqrt{2}}{2} \right) = -4\sqrt{2} \)
• \( b = 8 \times \left( -\frac{\sqrt{2}}{2} \right) = -4\sqrt{2} \)

So the rectangular form is \( -4\sqrt{2} - 4i\sqrt{2} \).

Modulus and Argument of Complex Numbers

Understanding modulus and argument is key to unlocking the power of complex numbers. The modulus, denoted as \(|z|\), measures the size or length of a complex number in the complex plane. You can find it by taking the square root of the sum of the squares of the real and imaginary parts \( |z| = \sqrt{a^2 + b^2} \).
The argument, denoted as \( \text{arg}(z) \), is the angle formed with the positive real axis. For a number expressed as \( a + bi \), the argument \( \theta \) can be found using the inverse tangent \( \theta = \tan^{-1}(\frac{b}{a}) \). This gives the direction of the complex number from the origin.
In the given exercise, the moduli of \( z_1 \) and \( z_2 \) are directly given as \( 4 \) and \( 2 \) respectively. For their product, the modulus is computed as \(|z_1 z_2| = 8\), and the argument is calculated by adding their individual arguments:

• \( \text{arg}(z_1) = 80^{\circ} \)
• \( \text{arg}(z_2) = 145^{\circ} \)
• The sum, \( 225^{\circ} \), is the argument of the product.

This streamlined approach helps in efficiently performing complex multiplication.