Problem 60 - Algebra and Trigonometry

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=\frac{4}{5}\left(\cos 25^{\circ}+i \sin 25^{\circ}\right)} \\ {z_{2}=\frac{1}{5}\left(\cos 155^{\circ}+i \sin 155^{\circ}\right)}\end{array} $$

Step-by-Step Solution

Answer

Product: \( \frac{4}{25}(\cos 180^\circ + i \sin 180^\circ) \); Quotient: \( 4(\cos(-130^\circ) + i \sin(-130^\circ)) \).

Step 1: Identify Polar Forms

We identify that both \(z_1\) and \(z_2\) are already in polar form. \(z_1\) has a magnitude of \(\frac{4}{5}\) and argument of \(25^\circ\). \(z_2\) has a magnitude of \(\frac{1}{5}\) and argument of \(155^\circ\).

Step 2: Product of Complex Numbers in Polar Form

To find the product \(z_1 z_2\), multiply the magnitudes and add the angles: \[ |z_1 z_2| = \frac{4}{5} \times \frac{1}{5} = \frac{4}{25} \] \[ \text{Argument of } z_1 z_2 = 25^\circ + 155^\circ = 180^\circ \] Thus, \(z_1 z_2 = \frac{4}{25}(\cos 180^\circ + i \sin 180^\circ)\).

Step 3: Quotient of Complex Numbers in Polar Form

To find the quotient \(\frac{z_1}{z_2}\), divide the magnitudes and subtract the angles: \[ \left| \frac{z_1}{z_2} \right| = \frac{4/5}{1/5} = 4 \] \[ \text{Argument of } \frac{z_1}{z_2} = 25^\circ - 155^\circ = -130^\circ \] So, \(\frac{z_1}{z_2} = 4(\cos(-130^\circ) + i \sin(-130^\circ))\).

Step 4: Express Results in Polar Form

Write the results of the product and quotient: \[ z_1 z_2 = \frac{4}{25}(\cos 180^\circ + i \sin 180^\circ) \] \[ \frac{z_1}{z_2} = 4(\cos(-130^\circ) + i \sin(-130^\circ)) \] These expressions are in polar form, with magnitude and angle identified in each case.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are usually written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In polar form, a complex number is expressed using its magnitude (or modulus) and its angle (or argument). This is usually represented as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle.
Representing complex numbers in polar form simplifies the process of multiplication and division. It breaks down each complex number into a format that highlights its length and direction.

• Magnitude \( r \) can be found using the formula \( r = \sqrt{a^2 + b^2} \).
• Angle \( \theta \) is the angle with the positive x-axis, calculated as \( \theta = \tan^{-1}(b/a) \).

Polar form is particularly useful in fields such as engineering and physics, where these operations are common.

Product of Complex Numbers

When multiplying complex numbers in polar form, the operation becomes quite straightforward. You simply need to multiply their magnitudes and add their angles.
For example, if you have two complex numbers \( z_1 = r_1(\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \), their product \( z_1 z_2 \) can be expressed in polar form.

• Multiply the magnitudes: \( |z_1 z_2| = r_1 \times r_2 \).
• Add the angles: \( \text{angle of } z_1 z_2 = \theta_1 + \theta_2 \).

This makes the entire multiplication process simpler, avoiding the need for expanding expressions and simplifying terms like in rectangular form. For instance, in the exercise, we found the product \( z_1 z_2 = \frac{4}{25}(\cos 180^\circ + i \sin 180^\circ) \), by applying these straightforward steps.

Quotient of Complex Numbers

Dividing complex numbers in polar form is similarly simplified by this representation. To find the quotient, we divide the magnitudes and subtract the angles. This eliminates complexities associated with division in rectangular form.
Given two polar form complex numbers, \( z_1 = r_1(\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \), the quotient \( \frac{z_1}{z_2} \) is calculated in the following way:

• Divide the magnitudes: \( \left| \frac{z_1}{z_2} \right| = \frac{r_1}{r_2} \).
• Subtract the angles: \( \text{angle of } \frac{z_1}{z_2} = \theta_1 - \theta_2 \).

This method simplifies calculations, as illustrated in our example: the quotient \( \frac{z_1}{z_2} = 4(\cos(-130^\circ) + i \sin(-130^\circ)) \) was determined quickly using these straightforward steps. Polar form streamlines these operations by focusing on the nature of the complex numbers as rotations and scalings in the complex plane.