Problem 55 - Algebra and Trigonometry

Question

Find the 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

The product is \(-15i\); the quotient in polar form is \(\frac{3}{5}(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})\).

Step 1: Convert to Polar Form

Recognize that the given complex numbers are already in polar form, where each complex number is represented as \( z = r(\cos \theta + i \sin \theta) \). Thus, for \( z_1 \), \( r_1 = 3 \) and \( \theta_1 = \frac{\pi}{6} \). For \( z_2 \), \( r_2 = 5 \) and \( \theta_2 = \frac{4\pi}{3} \).

Step 2: Calculate the Product z_1 z_2

To find the product of two complex numbers in polar form: \( z_1 z_2 = r_1 r_2 [ \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) ] \). Substitute the values: \[ z_1 z_2 = 3 \times 5 \left( \cos\left(\frac{\pi}{6} + \frac{4\pi}{3}\right) + i \sin\left(\frac{\pi}{6} + \frac{4\pi}{3}\right) \right) \]. \[ z_1 z_2 = 15 \left( \cos\left(\frac{9\pi}{6}\right) + i \sin\left(\frac{9\pi}{6}\right) \right) \]. Simplify: \( \frac{9\pi}{6} = \frac{3\pi}{2} \). \[ z_1 z_2 = 15 \left( \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right) \right) \].

Step 3: Simplify the Product Result

The angle \( \frac{3\pi}{2} \) corresponds to "-i" on the unit circle, so: \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \). Thus \[ z_1 z_2 = 15 \times (0 - i) = -15i \].

Step 4: Calculate the Quotient z_1 / z_2

For the quotient: \( \frac{z_1}{z_2} = \frac{r_1}{r_2} [ \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) ] \). Substitute the given values: \[ \frac{z_1}{z_2} = \frac{3}{5} \left( \cos\left(\frac{\pi}{6} - \frac{4\pi}{3}\right) + i \sin\left(\frac{\pi}{6} - \frac{4\pi}{3}\right) \right) \]. Calculate the angle: \( \frac{\pi}{6} - \frac{4\pi}{3} = \frac{\pi}{6} - \frac{8\pi}{6} = -\frac{7\pi}{6} \).

Step 5: Simplify the Quotient Result

Firstly, simplify the angle \( -\frac{7\pi}{6} \) by adding \( 2\pi \), giving \( \frac{5\pi}{6} \). Then, express the quotient: \[ \frac{z_1}{z_2} = \frac{3}{5} \left( \cos\left(\frac{5\pi}{6}\right) + i \sin\left(\frac{5\pi}{6}\right) \right) \]. The angle \( \frac{5\pi}{6} \) corresponds to \( \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \) on the unit circle.

Key Concepts

Product of Complex Numbers

Understanding the product of complex numbers in polar form is quite straightforward. In polar form, each complex number is expressed as \( z = r (\cos \theta + i \sin \theta) \). Here, \( r \) represents the modulus (or length) of the complex number, while \( \theta \) is the argument (or angle).

When calculating the product of two complex numbers, \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), it is essential to remember that the moduli multiply while the arguments add. This is summarized in the equation:

• \( z_1 z_2 = r_1 r_2 [ \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) ] \)

To illustrate, substituting the values from the original exercise, \( z_1 = 3(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) \) and \( z_2 = 5(\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}) \), gives:
\[ z_1 z_2 = 3 \times 5 \left( \cos\left(\frac{\pi}{6} + \frac{4\pi}{3}\right) + i \sin\left(\frac{\pi}{6} + \frac{4\pi}{3}\right) \right) \]
Simplifying \( \frac{\pi}{6} + \frac{4\pi}{3} \) results in \( \frac{9\pi}{6} = \frac{3\pi}{2} \). Hence, the product is \( z_1 z_2 = 15 (0 - i) = -15i \). It's helpful to visualize this as a rotation and scaling in a complex plane.

Quotient of Complex Numbers

Calculating the quotient of complex numbers in polar form requires a similar approach to finding the product. However, instead of multiplying the moduli, you divide them, and instead of adding the arguments, you subtract them.

To find the quotient \( \frac{z_1}{z_2} \) of two complex numbers given in polar form \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), the formula used is:

• \( \frac{z_1}{z_2} = \frac{r_1}{r_2} [ \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) ] \)

Using the values \( r_1 = 3 \), \( \theta_1 = \frac{\pi}{6} \), \( r_2 = 5 \), and \( \theta_2 = \frac{4\pi}{3} \) as seen in the original exercise, the equation becomes:
\[ \frac{z_1}{z_2} = \frac{3}{5} \left( \cos\left(\frac{\pi}{6} - \frac{4\pi}{3}\right) + i \sin\left(\frac{\pi}{6} - \frac{4\pi}{3}\right) \right) \]
Simplifying \( \frac{\pi}{6} - \frac{4\pi}{3} \) results in \( -\frac{7\pi}{6} \), which can further be adjusted by adding \( 2\pi \) to be \( \frac{5\pi}{6} \). Thus, the result is:
\[ \frac{z_1}{z_2} = \frac{3}{5} \left( -\frac{\sqrt{3}}{2} + \frac{1}{2} i \right) \].
This calculation elegantly shows the division of one complex vector by another, demonstrating how they relate in terms of magnitude and direction.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers, often called polar form, is a powerful way to express complex numbers. This expression uses the identity \( z = r (\cos \theta + i \sin \theta) \), where \( r \) is the modulus of the complex number and \( \theta \) is its argument.

The modulus \( r \) is calculated as \( |z| = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are real and imaginary parts, respectively. It represents the distance of the complex number from the origin in the complex plane. The argument \( \theta \) is the angle made with the positive real axis, usually calculated using the arctan function: \( \theta = \tan^{-1}(\frac{y}{x}) \).

The trigonometric form simplifies the multiplication and division of complex numbers, as discussed previously. Adding a geometric perspective, you can think of it as rotating a vector in the plane by angle \( \theta \) and stretching it to length \( r \). This form is particularly elegant for visualizations, analysis of periodic functions, and solving complex equations, allowing direct manipulation of the angle and magnitude of complex numbers.