Problem 56 - 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. $$ \begin{array}{l}{z_{1}=\frac{4}{5}\left(\cos 25^{\circ}+i \sin 25^{\circ}\right)} \\ {z_{2}=\frac{1}{5}\left(\cos 155^{\circ}+i \sin 155^{\circ}\right)}\end{array} $$

Step-by-Step Solution

Answer

Product: \(-\frac{4}{25}\), Quotient in polar form: \(4 \text{cis} 230^{\circ}\).

Step 1: Express Each Complex Number in Polar Form

Given that complex numbers are described in the form \(z = r(\cos\theta + i\sin\theta)\), we recognize this as the polar form already. Hence, for \(z_1 = \frac{4}{5}(\cos 25^{\circ} + i \sin 25^{\circ})\), the modulus is \(r_1 = \frac{4}{5}\) and the argument is \(\theta_1 = 25^{\circ}\). Similarly, for \(z_2 = \frac{1}{5}(\cos 155^{\circ} + i \sin 155^{\circ})\), the modulus is \(r_2 = \frac{1}{5}\) and the argument is \(\theta_2 = 155^{\circ}\).

Step 2: Calculate the Product of the Complex Numbers

The product \(z_1 z_2\) in polar form is given by multiplying the moduli and adding the arguments. Thus, \[ z_1 z_2 = \left(\frac{4}{5} \times \frac{1}{5}\right) \left(\cos(25^{\circ} + 155^{\circ}) + i\sin(25^{\circ} + 155^{\circ})\right) \].So, \[ z_1 z_2 = \frac{4}{25} \left(\cos(180^{\circ}) + i\sin(180^{\circ})\right) \].Since \(\cos(180^{\circ}) = -1\) and \(\sin(180^{\circ}) = 0\), this simplifies to \[ z_1 z_2 = -\frac{4}{25} \].

Step 3: Calculate the Quotient of the Complex Numbers

The quotient \(\frac{z_1}{z_2}\) in polar form is found by dividing the moduli and subtracting the arguments:\[ \frac{z_1}{z_2} = \left(\frac{4/5}{1/5}\right) \left(\cos(25^{\circ} - 155^{\circ}) + i\sin(25^{\circ} - 155^{\circ})\right) \].This gives\[ \frac{z_1}{z_2} = 4 \left(\cos(-130^{\circ}) + i\sin(-130^{\circ})\right) \],which is also expressed as \[ 4 \left(\cos(230^{\circ}) + i\sin(230^{\circ})\right) \].

Key Concepts

Polar Form

Complex numbers can be represented in different forms, and one of the most intuitive is the polar form. This form expresses complex numbers using a magnitude (or modulus) and an angle (or argument) from the positive real axis. In polar form, a complex number is written as:

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

Here, \( r \) is the modulus and \( \theta \) is the argument. The modulus represents the distance of the complex number from the origin in the complex plane, while the argument is the counterclockwise angle from the positive real axis to the line representing the complex number.
By expressing complex numbers in polar form, operations like multiplication and division become significantly easier. This is because the modulus and argument simplify the process, allowing the operations to be reduced to basic arithmetic, specifically multiplication/division of moduli and addition/subtraction of arguments.

Complex Multiplication

Multiplying complex numbers in polar form is straightforward and convenient. The modulus of the product is the product of the moduli of the original numbers, and the argument of the product is the sum of the arguments.
Given two complex numbers:

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

Their product \( z_1 z_2 \) is represented in polar form as:

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

For example, if \( z_1 = \frac{4}{5}(\cos 25^{\circ} + i \sin 25^{\circ}) \) and \( z_2 = \frac{1}{5}(\cos 155^{\circ} + i \sin 155^{\circ}) \), the product is:

• Modulus: \( \frac{4}{5} \times \frac{1}{5} = \frac{4}{25} \)
• Argument: \( 25^{\circ} + 155^{\circ} = 180^{\circ} \)

Thus, the product is \( z_1 z_2 = \frac{4}{25}(\cos 180^{\circ} + i \sin 180^{\circ}) \). Since \( \cos 180^{\circ} = -1 \) and \( \sin 180^{\circ} = 0 \), this simplifies to \( -\frac{4}{25} \).

Complex Division

Dividing complex numbers is also more convenient in polar form. To find the quotient, divide the modulus of the first complex number by the modulus of the second, and subtract the argument of the second from the first. Consider complex numbers:

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

Their quotient \( \frac{z_1}{z_2} \) is given by:

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

For the example with \( z_1 = \frac{4}{5}(\cos 25^{\circ} + i \sin 25^{\circ}) \) and \( z_2 = \frac{1}{5}(\cos 155^{\circ} + i \sin 155^{\circ}) \), the division results in:

• Modulus: \( \frac{4/5}{1/5} = 4 \)
• Argument: \( 25^{\circ} - 155^{\circ} = -130^{\circ} \)

This is often expressed using a positive angle by adding 360 degrees, giving \( 230^{\circ} \). Therefore, the quotient is expressed as \( 4(\cos 230^{\circ} + i \sin 230^{\circ}) \).

Modulus and Argument

The modulus and argument are key components of a complex number in polar form. Understanding these components is crucial for easily performing complex arithmetic.

Modulus

The modulus \( r \) of a complex number \( z = x + yi \) is calculated as \( r = \sqrt{x^2 + y^2} \). It represents the length of the vector (or point) in the complex plane from the origin. In polar expressions, the modulus scales the vector's length, indicating the complex number's magnitude.

Argument

The argument \( \theta \) is the angle formed with the positive real axis. It gives the direction of the vector line in the complex plane. Mathematically, it is often computed using the inverse tangent function: \( \theta = \tan^{-1}\left( \frac{y}{x} \right) \). To let \( \theta \) be in the desired quadrant according to the position of \( x \) and \( y \), adjustments may be needed. Argues in degrees or radians, and each leads to equivalent representation after a full rotation (360 degrees or \( 2\pi \) radians).
Every complex number's distinct coordinate in the complex plane can be captured by its unique modulus and argument, providing a powerful and simple way to describe its behavior and interactions with other complex numbers.