Problem 6 - Precalculus

Question

Find the product \(z_{1} z_{2}\) and express it in rectangular form. $$z_{1}=3\left(\cos 190^{\circ}+i \sin 190^{\circ}\right) \text { and } z_{2}=5\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)$$

Step-by-Step Solution

Answer

The product is \(-15i\) in rectangular form.

Step 1: Express Each Number in Exponential Form

To convert a complex number from trigonometric form to exponential form, use the formula \( z = r e^{i \theta} \), where \( r \) is the modulus and \( \theta \) is the argument.For \( z_1 = 3(\cos 190^{\circ} + i \sin 190^{\circ}) \), the exponential form is \( z_1 = 3 e^{i 190^{\circ}} \).For \( z_2 = 5(\cos 80^{\circ} + i \sin 80^{\circ}) \), the exponential form is \( z_2 = 5 e^{i 80^{\circ}} \).

Step 2: Multiply the Moduli and Add the Arguments

The product of two complex numbers in exponential form is given by \( r_1 e^{i \theta_1} \times r_2 e^{i \theta_2} = (r_1 r_2) e^{i(\theta_1 + \theta_2)} \).Multiply the moduli: \( 3 \times 5 = 15 \).Add the arguments: \( 190^{\circ} + 80^{\circ} = 270^{\circ} \).Therefore, the product in exponential form is \( 15 e^{i 270^{\circ}} \).

Step 3: Express the Product in Trigonometric Form

Convert the exponential form back to trigonometric form using \( z = r (\cos \theta + i \sin \theta) \).The product is \( 15 (\cos 270^{\circ} + i \sin 270^{\circ}) \).From the unit circle, \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \).Thus, the product in trigonometric form is \( 15 (0 + i (-1)) = -15i \).

Step 4: Express the Product in Rectangular Form

In rectangular form, a complex number is written as \( a + bi \). From the previous step:The rectangular form is \( 0 - 15i \), or simply \( -15i \).

Key Concepts

Trigonometric Form

Complex numbers can be represented in various forms, with the trigonometric form being one of the most intuitive for visualizing the number's properties on the complex plane. This form expresses a complex number using its modulus (or length) and an angle known as the argument, relative to the positive x-axis.
To write a complex number in trigonometric form, you use the formula:

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

where:

• \( r \) is the modulus, representing the distance from the origin to the point on the complex plane.
• \( \theta \) is the argument, indicating the angle formed with the positive x-axis.

For example, in the given exercise, the numbers \( z_{1} \) and \( z_{2} \) are written in trigonometric form, helping us easily visualize and compute further operations such as multiplication or division.

Exponential Form

The exponential form of a complex number combines the trigonometric form with the mathematical elegance of Euler's formula. This form is particularly handy in more advanced calculations, such as powers and roots, due to its simplicity and ease in handling multiplicative operations.
Euler's formula tells us:

• \( e^{i \theta} = \cos \theta + i \sin \theta \)

So, a complex number \( z \) can also be expressed as:

• \( z = r e^{i \theta} \)

This transformation into exponential form retains the modulus \( r \) and the angle \( \theta \) as in the trigonometric form, with \( e^{i \theta} \) encapsulating the angle information.
In the solved exercise, converting \( z_{1} \) and \( z_{2} \) into exponential form allows an easier multiplication process. The product of two complex numbers \( z_{1} \) and \( z_{2} \) becomes a straightforward calculation as you multiply the moduli and add the angles, resulting in a neat exponential product.

Rectangular Form

The rectangular form, also known as the standard or Cartesian form, is the most straightforward representation of a complex number. It provides an immediate way to understand the real and imaginary components of a number.
In rectangular form, a complex number is expressed as:

• \( z = a + bi \)

where:

• \( a \) is the real part.
• \( b \) is the imaginary part.

This form is invaluable for addition and subtraction operations, as it allows these to be done component-wise.
In the exercise, after performing the multiplication in exponential and trigonometric forms, the final product was expressed in rectangular form as \( -15i \). Here, the real part is zero, indicating that the number lies entirely along the imaginary axis on the complex plane.