Problem 51 - Algebra and Trigonometry

Question

\(49-56\) me product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ z_{1}=3\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right), \quad z_{2}=5\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right) $$

Step-by-Step Solution

Answer

Product: \(15\text{cis}\left(\frac{3\pi}{2}\right)\); Quotient: \(\frac{3}{5}\text{cis}\left(-\frac{7\pi}{6}\right)\).

Step 1: Convert to Polar Form

Both numbers are already given in polar form as \(z_1 = 3\text{cis}\left(\frac{\pi}{6}\right)\) and \(z_2 = 5\text{cis}\left(\frac{4\pi}{3}\right)\), where \(\text{cis}\theta = \cos\theta + i\sin\theta\). Thus, \( z_1 = 3\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\) and \( z_2 = 5\left(\cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3}\right)\) are already in polar form.

Step 2: Find the Product

The product of two complex numbers in polar form \(z_1 = r_1\text{cis}\theta_1\) and \(z_2 = r_2\text{cis}\theta_2\) is given by \(z_1z_2 = r_1r_2\text{cis}(\theta_1 + \theta_2)\). For \(z_1 = 3\text{cis}\left(\frac{\pi}{6}\right)\) and \(z_2 = 5\text{cis}\left(\frac{4\pi}{3}\right)\), we have: \[z_1 z_2 = 3 \times 5 \times \text{cis}\left(\frac{\pi}{6} + \frac{4\pi}{3}\right)\ = 15\text{cis}\left(\frac{\pi}{6} + \frac{4\pi}{3}\right)\ = 15\text{cis}\left(\frac{9\pi}{6}\right)\ = 15\text{cis}\left(\frac{3\pi}{2}\right).\]

Step 3: Find the Quotient

The quotient of two complex numbers in polar form \(z_1 = r_1\text{cis}\theta_1\) and \(z_2 = r_2\text{cis}\theta_2\) is given by \(\frac{z_1}{z_2} = \frac{r_1}{r_2}\text{cis}(\theta_1 - \theta_2)\). For \(z_1 = 3\text{cis}\left(\frac{\pi}{6}\right)\) and \(z_2 = 5\text{cis}\left(\frac{4\pi}{3}\right)\), we have: \[ \frac{z_1}{z_2} = \frac{3}{5}\text{cis}\left(\frac{\pi}{6} - \frac{4\pi}{3}\right) = \frac{3}{5}\text{cis}\left(\frac{\pi}{6} - \frac{8\pi}{6}\right) = \frac{3}{5}\text{cis}\left(-\frac{7\pi}{6}\right). \]

Key Concepts

Polar Form

Complex numbers can be expressed in different ways, and one of the most insightful forms is the polar form. It represents a complex number through its magnitude and angle.

In polar form, a complex number is written as \( z = r(\cos \theta + i \sin \theta) \), often abbreviated as \( z = r \text{cis} \theta \). Here, \( r \) is the magnitude, and \( \theta \) is the angle (or argument) measured counterclockwise from the positive x-axis.

Using the polar form can be particularly helpful in performing multiplication and division of complex numbers due to the elegant simplification it provides.

• Magnitude \( r \): Calculated as \( \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the real and imaginary components of the complex number.
• Argument \( \theta \): Found using the inverse tangent function \( \tan^{-1}\left(\frac{y}{x}\right) \).

By converting complex numbers into polar form, calculations involving multiplication or division become straightforward.

Product of Complex Numbers

When multiplying two complex numbers in polar form, the operation becomes significantly simpler compared to using the standard form. Given two complex numbers in polar form \( z_1 = r_1 \text{cis} \theta_1 \) and \( z_2 = r_2 \text{cis} \theta_2 \), the product is derived by multiplying the magnitudes and adding the angles:

\[z_1 z_2 = r_1 r_2 \text{cis}(\theta_1 + \theta_2)\]

This formula shows the power of polar form – what could involve extensive algebraic manipulation in rectangular form is now a simple task of handling magnitudes and angles.

• Multiply magnitudes: Combine the magnitudes \( r_1 \) and \( r_2 \) by simple multiplication.
• Add angles: The new angle is the sum of the original angles \( \theta_1 \) and \( \theta_2 \).

This approach makes it clear why polar form is advantageous for multiplication, transforming complex multiplication into mere arithmetic.

Quotient of Complex Numbers

Just as with multiplication, dividing complex numbers in polar form simplifies the process significantly. Consider two complex numbers \( z_1 = r_1 \text{cis} \theta_1 \) and \( z_2 = r_2 \text{cis} \theta_2 \). The quotient is calculated as follows:

\[\frac{z_1}{z_2} = \frac{r_1}{r_2} \text{cis}(\theta_1 - \theta_2)\]

This formula underscores the elegance of polar form in division:

• Divide magnitudes: Simply divide the magnitudes \( r_1 \) by \( r_2 \).
• Subtract angles: The new angle is the difference between \( \theta_1 \) and \( \theta_2 \).

By transforming division into operations on magnitudes and angles, complex number division in polar form strips away the complexity, streamlining calculations and improving clarity.