Problem 52 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\frac{\cos 120^{\circ}+i \sin 120^{\circ}}{2\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)}$$

Step-by-Step Solution

Answer

The result of the operation is \( 0.5(\cos 80^{\circ} + i \sin 80^{\circ}) \).

Step 1: Convert to Polar Form

Write each of the given complex numbers in polar form. The general polar form of a complex number is \( r(\cos \theta + i \sin \theta) \) so the given complex numbers can be written as: The numerator as \( 1(\cos 120^{\circ} + i \sin 120^{\circ}) \) and the denominator as \( 2(\cos 40^{\circ} + i \sin 40^{\circ}) \).

Step 2: Apply the Division Operation

The division operation in polar form is done as follows: If we have two complex numbers \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \), then their division is \( (r_1/r_2)(\cos(\theta_1-\theta_2) + i \sin(\theta_1-\theta_2)) \). Applying this to the given numbers, we get: \( \frac{1}{2}(\cos(120^{\circ}-40^{\circ}) + i \sin(120^{\circ}-40^{\circ})) \).

Step 3: Simplify

We now need to simplify the expression. The first part is simple division, \( \frac{1}{2} = 0.5 \). The other part is subtraction of two angles, \( 120^{\circ}-40^{\circ} = 80^{\circ} \). Therefore, the simplified complex number in trigonometric form is \( 0.5(\cos 80^{\circ} + i \sin 80^{\circ}) \).

Key Concepts

Polar Form

Complex numbers can be expressed in different forms, and one of the most intuitive ways is through the **polar form**. Imagine your complex number as a vector in the complex plane. To describe this vector, you need two key pieces of information: its length (also known as the magnitude) and the angle it makes with the positive real axis.

In polar form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle. This angle is often referred to as the argument of the complex number.

• **Magnitude \( r \)**: This is the length of the vector from the origin to the point representing the complex number.
• **Argument \( \theta \)**: The direction of the vector is indicated by \( \theta \), typically measured in degrees or radians, counterclockwise from the positive x-axis.

The beauty of the polar form is that it easily translates complex number operations, such as multiplication and division, into operations with real numbers.

Division of Complex Numbers

Dividing complex numbers might seem tricky at first, but it's quite straightforward once you convert them into polar form. Using polar form simplifies the operation significantly.

To divide two complex numbers in polar form, you divide their magnitudes and subtract their angles. Consider two complex numbers \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \).

The division operation can be written as:

\[\frac{r_1}{r_2} \left( \cos(\theta_1-\theta_2) + i \sin(\theta_1-\theta_2) \right)\]This gives you the magnitude and direction of the resulting complex number.

• The **magnitude** of the resulting complex number is \( \frac{r_1}{r_2} \).
• The **angle** or argument of the resulting number is the difference \( \theta_1 - \theta_2 \).

This method makes it much easier to handle complex divisions without getting lost in algebraic manipulations.

Trigonometric Form

The **trigonometric form** of a complex number is closely linked to its polar form. Essentially, it is another way of expressing the same information: the magnitude and angle of a complex number.

The trigonometric form uses the same expression, \( r(\cos \theta + i \sin \theta) \), denoting the magnitude and direction as before. This form is extremely useful, especially when you perform operations like multiplication, division, or finding powers and roots of complex numbers.

Here are some features and benefits of using the trigonometric form:

• **Ease of operations**: Multiplication and division become operations on real numbers, i.e., multiplying/dividing magnitudes and adding/subtracting angles.
• **Clarified visualization**: The trigonometric form helps visualize the behavior of complex numbers in the plane, showing how they rotate and scale.
• **Connection to Euler's Formula**: You can relate the trigonometric form to the exponential form through Euler's formula: \( e^{i\theta} = \cos \theta + i\sin \theta \).

The trigonometric form makes the idea of a complex number more tangible and is incredibly helpful for calculations that involve complex number interactions.