Problem 57 - Algebra and Trigonometry

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)} \\\ {z_{2}=2\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)}\end{array} $$

Step-by-Step Solution

Answer

Product: \(8(\cos 150^\circ + i \sin 150^\circ)\); Quotient: \(2(\cos 90^\circ + i \sin 90^\circ)\).

Step 1: Understanding the Problem

We are given two complex numbers in polar form: \( z_1 = 4(\cos 120^\circ + i \sin 120^\circ) \) and \( z_2 = 2(\cos 30^\circ + i \sin 30^\circ) \). We need to find the product \( z_1 z_2 \) and the quotient \( \frac{z_1}{z_2} \), both in polar form.

Step 1: Find the Product

To find \( z_1 z_2 \), multiply the magnitudes and add the angles: \( |z_1| \cdot |z_2| = 4 \cdot 2 = 8 \). The angle is \( 120^\circ + 30^\circ = 150^\circ \). Thus, the product is \( 8(\cos 150^\circ + i \sin 150^\circ) \).

Step 2: Find the Quotient

To find \( \frac{z_1}{z_2} \), divide the magnitudes and subtract the angles: \( \frac{|z_1|}{|z_2|} = \frac{4}{2} = 2 \). The angle is \( 120^\circ - 30^\circ = 90^\circ \). Thus, the quotient is \( 2(\cos 90^\circ + i \sin 90^\circ) \).

Step 4: Solution in Polar Form

The product \( z_1 z_2 \) in polar form is \( 8(\cos 150^\circ + i \sin 150^\circ) \) and the quotient \( \frac{z_1}{z_2} \) in polar form is \( 2(\cos 90^\circ + i \sin 90^\circ) \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \).

A complex number can also be expressed in polar form, which is especially useful for multiplication and division. The polar form is \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle). Writing a complex number in this form makes it easy to interpret geometrically and to perform operations on complex numbers.

Polar coordinates are an incredibly helpful way of working with complex numbers, especially when considering their multiplication and division, because it simplifies the operations into simpler arithmetic on the magnitudes and angles.

Product and Quotient of Complex Numbers

The product and quotient of complex numbers become much simpler to handle when these numbers are in polar form.

• **Product**: To find the product of two complex numbers \( z_1 \) and \( z_2 \) given in polar form as \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \), you multiply their magnitudes and add their angles:\[ |z_1 z_2| = r_1r_2 \]\[ \text{Angle of } z_1 z_2 = \theta_1 + \theta_2 \]This results in the product \( r_1 r_2(\cos(\theta_1+\theta_2) + i \sin(\theta_1+\theta_2)) \).
• **Quotient**:To find the quotient \( \frac{z_1}{z_2} \) of two complex numbers, you divide their magnitudes and subtract their angles:\[ |\frac{z_1}{z_2}| = \frac{r_1}{r_2} \]\[ \text{Angle of } \frac{z_1}{z_2} = \theta_1 - \theta_2 \]This results in the quotient \( \frac{r_1}{r_2}(\cos(\theta_1-\theta_2) + i \sin(\theta_1-\theta_2)) \).

Simplifying these operations becomes very handy when dealing with complex numbers in practical applications.

de Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number analysis that links complex numbers with trigonometry in a strategic manner. This theorem states:If \( z = r(\cos \theta + i \sin \theta) \) is a complex number in polar form and \( n \) is any integer, then:\[(z)^n = r^n(\cos(n\theta) + i \sin(n\theta))\]This formula is exceptionally useful for computing powers and roots of complex numbers without reverting to the standard rectangular form. Applying de Moivre's Theorem can simplify calculations significantly, making it a favorite for dealing with complex exponential growth and solving power equations. It highlights the beauty of polar coordinates by demonstrating how elegantly and efficiently they allow for operations on complex numbers, especially in exponential contexts.