Problem 2 - Precalculus

Question

Find the product \(z_{1} z_{2}\) and express it in rectangular form. $$z_{1}=2\left(\cos 100^{\circ}+i \sin 100^{\circ}\right) \text { and } z_{2}=5\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)$$

Step-by-Step Solution

Answer

The product is \(-5\sqrt{3} + 5i\).

Step 1: Understand the given complex numbers in polar form

The given complex numbers are in polar form, where each is represented as \( z = r(\cos \theta + i \sin \theta) \). Here, \( z_{1} = 2(\cos 100^{\circ} + i \sin 100^{\circ}) \) and \( z_{2} = 5(\cos 50^{\circ} + i \sin 50^{\circ}) \). The magnitude of \( z_{1} \) is 2, and \( z_{2} \) is 5.

Step 2: Calculate the product's magnitude and angle

When multiplying two complex numbers in polar form, the magnitudes are multiplied and the angles are added. Therefore, the magnitude of \( z_{1} z_{2} \) is \( 2 \times 5 = 10 \) and the angle is \( 100^{\circ} + 50^{\circ} = 150^{\circ} \).

Step 3: Express the product in polar form

Substitute the calculated magnitude and angle into the polar form: \( z_{1}z_{2} = 10(\cos 150^{\circ} + i \sin 150^{\circ}) \).

Step 4: Convert to rectangular form using known values for trigonometric functions

Now, we translate the polar form into rectangular form using the exact trigonometric values: \( \cos 150^{\circ} = -\frac{\sqrt{3}}{2} \) and \( \sin 150^{\circ} = \frac{1}{2} \). Substitute these into the expression: \( z_{1}z_{2} = 10\left(-\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2}\right) \).

Step 5: Simplify the expression to get the rectangular form

Distribute the 10 across the terms: \( z_{1}z_{2} = 10 \times -\frac{\sqrt{3}}{2} + 10 \times i \cdot \frac{1}{2} = -5\sqrt{3} + 5i \).

Step 6: Final expression in rectangular form

The product \( z_{1}z_{2} \) in rectangular form is \( -5\sqrt{3} + 5i \).

Key Concepts

Polar Form

Complex numbers can be represented in different forms, and one of the most common is the polar form. In this form, a complex number is expressed as \( z = r(\cos \theta + i \sin \theta) \), where:

• \( r \) (magnitude or modulus) is the distance from the origin to the point in the complex plane.
• \( \theta \) (angle or argument) is the angle formed with the positive x-axis, measured in degrees or radians.

Polar form is particularly useful for multiplication and division of complex numbers.
Let's take a look at our example. The two complex numbers given are:

• \( z_{1}=2(\cos 100^{\circ}+i \sin 100^{\circ}) \)
• \( z_{2}=5(\cos 50^{\circ}+i \sin 50^{\circ}) \)

Here, \( z_{1} \) and \( z_{2} \) are clearly stated in polar form.
For \( z_{1} \), the magnitude is 2 and the angle is \( 100^{\circ} \).
For \( z_{2} \), the magnitude is 5 and the angle is \( 50^{\circ} \).
Understanding this form makes it easier to perform operations like multiplication.

Rectangular Form

The rectangular form of a complex number represents it in terms of real and imaginary components, written as \( a + bi \).
Here:

• \( a \) is the real part.
• \( b \) is the imaginary part, multiplied by \( i \), where \( i \) is the imaginary unit \( \sqrt{-1} \).

Every complex number has an equivalent rectangular form, which is very intuitive and easy to visualize on the complex plane.
To convert from polar to rectangular form, we use the trigonometric identities:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

For the product \( z_{1}z_{2} \) in our exercise, expressed in polar form as \( 10(\cos 150^{\circ} + i \sin 150^{\circ}) \),
converting to rectangular form involves substituting the known values:

• \( \cos 150^{\circ} = -\frac{\sqrt{3}}{2} \)
• \( \sin 150^{\circ} = \frac{1}{2} \)

The calculated rectangular form becomes:

• \( z_{1}z_{2} = 10\left(-\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2}\right) = -5\sqrt{3} + 5i \)

This form provides a clearer picture of the complex number's real and imaginary parts.

Multiplication of Complex Numbers

Multiplying complex numbers becomes straightforward when they are in polar form.
The process involves two primary steps: combining the magnitudes and adding the angles. Given two complex numbers \( z_{1} = r_{1}(\cos \theta_{1} + i \sin \theta_{1}) \) and \( z_{2} = r_{2}(\cos \theta_{2} + i \sin \theta_{2}) \), the product is:

• Magnitude: \( r = r_{1} \times r_{2} \)
• Angle: \( \theta = \theta_{1} + \theta_{2} \)

In our exercise:

• \( z_{1} \) has a magnitude of 2 and angle \( 100^{\circ} \).
• \( z_{2} \) has a magnitude of 5 and angle \( 50^{\circ} \).

Thus, the product\'s magnitude is \( 2 \times 5 = 10 \), and its angle is \( 100^{\circ} + 50^{\circ} = 150^{\circ} \).
This polar form \( 10(\cos 150^{\circ} + i \sin 150^{\circ}) \) simplifies the computation.
It always leads to the correct rectangular form when needed, allowing us to confidently switch between these representations.