Problem 59 - Algebra and Trigonometry

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=4\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)} \\\ {z_{2}=25\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}\end{array} $$

Step-by-Step Solution

Answer

Product: \( 100(\cos 350^\circ + i \sin 350^\circ) \), Quotient: \( \frac{4}{25}(\cos 50^\circ + i \sin 50^\circ) \).

Step 1: Understand the Problem

We need to find the product \( z_1 z_2 \) and the quotient \( \frac{z_1}{z_2} \), and express each result in polar form. The given complex numbers are \( z_1 = 4(\cos 200^\circ + i \sin 200^\circ) \) and \( z_2 = 25(\cos 150^\circ + i \sin 150^\circ) \).

Step 2: Finding the Product \( z_1 z_2 \)

Using the formula for multiplying complex numbers in polar form, \( z_1 z_2 = |z_1||z_2|(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)) \). Here, \(|z_1| = 4\) and \(|z_2| = 25\), and their arguments \(\theta_1 = 200^\circ\) and \(\theta_2 = 150^\circ\). So, \( z_1 z_2 = 4 \times 25 (\cos(200^\circ + 150^\circ) + i\sin(200^\circ + 150^\circ)) = 100 (\cos(350^\circ) + i \sin(350^\circ)) \).

Step 3: Finding the Quotient \( \frac{z_1}{z_2} \)

Using the formula for dividing complex numbers in polar form, \( \frac{z_1}{z_2} = \frac{|z_1|}{|z_2|}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)) \). Here, \(|z_1| = 4\) and \(|z_2| = 25\), and their arguments \(\theta_1 = 200^\circ\) and \(\theta_2 = 150^\circ\). So, \( \frac{z_1}{z_2} = \frac{4}{25}(\cos(200^\circ - 150^\circ) + i\sin(200^\circ - 150^\circ)) = \frac{4}{25}(\cos(50^\circ) + i \sin(50^\circ)) \).

Step 4: Express the Results in Polar Form

The product is \( 100(\cos 350^\circ + i \sin 350^\circ) \), and the quotient is \( \frac{4}{25}(\cos 50^\circ + i \sin 50^\circ) \).

Key Concepts

Multiplying Complex Numbers in Polar Form

Multiplying complex numbers in polar form is relatively straightforward. Each complex number has a magnitude (or modulus) and an angle (or argument). Given two complex numbers, the product is found by:

• Multiplying their magnitudes.
• Adding their angles.

For example, let's consider the complex numbers given in polar form: \[ z_1 = 4(\cos 200^\circ + i \sin 200^\circ) \]\[ z_2 = 25(\cos 150^\circ + i \sin 150^\circ) \]Their magnitudes are 4 and 25, respectively, and their angles are 200° and 150°. To multiply these:
1. Multiply the magnitudes: \( 4 \times 25 = 100 \).
2. Add the angles: \( 200^\circ + 150^\circ = 350^\circ \).

The resulting product is expressed in polar form as:\[ 100(\cos 350^\circ + i \sin 350^\circ) \] This shows how the multiplication simplifies in polar form, where the process involves basic arithmetic operations with magnitudes and angles.

Dividing Complex Numbers in Polar Form

Dividing complex numbers in polar form is similar to multiplication, but with a twist. For division, the steps involve:

• Dividing the magnitudes.
• Subtracting the angles.

Taking the same complex numbers from the previous example:\[ z_1 = 4(\cos 200^\circ + i \sin 200^\circ) \]\[ z_2 = 25(\cos 150^\circ + i \sin 150^\circ) \]The steps for division are:
1. Divide the magnitudes: \( \frac{4}{25} \).
2. Subtract the angles: \( 200^\circ - 150^\circ = 50^\circ \).

Expressing the result in polar form gives:\[ \frac{4}{25}(\cos 50^\circ + i \sin 50^\circ) \]By dividing the magnitudes and subtracting the angles, we simplify the division process of complex numbers in polar form. This makes it much more efficient than working in rectangular form.

Understanding Polar Coordinates in Complex Numbers

Complex numbers have a unique representation in the polar coordinate system. In polar form, a complex number is expressed as:

• A magnitude, which indicates its distance from the origin.
• An angle, which represents its direction from the positive real axis.

The generic format is:\[ r(\cos \theta + i \sin \theta) \]Here, \( r \) is the magnitude, and \( \theta \) is the angle or argument. Polar form is particularly handy for multiplication and division, as it transforms these operations into simple arithmetic involving magnitudes and angles.
The polar representation makes visualization and computation involving rotational aspects of complex numbers much easier. Rather than dealing with the more cumbersome rectangular form \( a + bi \), where \( a \) and \( b \) are real numbers, polar coordinates simplify calculations by focusing on geometric interpretations. This approach is invaluable in fields such as engineering and physics, where rotational motion and periodic phenomena are prevalent.