Problem 58 - Precalculus with Limits

Question

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{6(\cos\ 40^{\circ} + i\ \sin\ 40^{\circ})}{7(\cos\ 100^{\circ} + i\ \sin\ 100^{\circ})}\)

Step-by-Step Solution

Answer

The simplified form of the given complex number is \(\frac{3}{7} - i \frac{3\sqrt{3}}{7}\).

Step 1: Apply the Quotient Property

The quotient property of complex numbers in trigonometric form states that \(\frac{r_1(\cos θ_1 + i \sin θ_1)}{r_2(\cos θ_2 + i \sin θ_2)} = \frac{r_1}{r_2}(\cos(θ_1 - θ_2) + i \sin(θ_1 - θ_2))\). Applying this property, we have \(\frac{6(\cos\ 40^{\circ} + i\ \sin\ 40^{\circ})}{7(\cos\ 100^{\circ} + i\ \sin\ 100^{\circ})} = \frac{6}{7}(\cos(40^{\circ}-100^{\circ}) + i\sin(40^{\circ}-100^{\circ}))\)

Step 2: Simplify Trigonometric Expressions

Now simplify the trigonometric expressions. The expression becomes: \(\frac{6}{7}(\cos(-60^{\circ}) + i\sin(-60^{\circ}))\). The cosine function is even, so \cos(-60^{\circ}) = \cos(60^{\circ}), and the sine function is odd, so \sin(-60^{\circ}) = -\sin(60^{\circ}). Thus, the expression can be written as \(\frac{6}{7}(\cos(60^{\circ}) - i\sin(60^{\circ}))\)

Step 3: Evaluate Trigonometric Values

Evaluate \cos(60^{\circ}) and \sin(60^{\circ}) to get their numerical values. The expression becomes: \(\frac{6}{7}(\frac{1}{2} - i \frac{\sqrt{3}}{2})\)

Step 4: Simplify The Expression

Simplify the expression further by multiplying the real and imaginary parts by \(\frac{6}{7}\). We have \(\frac{3}{7} - i \frac{3\sqrt{3}}{7}\)

Key Concepts

Quotient Property

The quotient property is a powerful tool when working with complex numbers in trigonometric form, also known as polar form. This property allows us to divide two complex numbers by simplifying the operation as a subtraction of angles and a division of magnitudes.

• When you have two complex numbers, the formula reads: \[ \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \]
• This means you divide the magnitudes \( r_1 \) and \( r_2 \), and subtract the angles \( \theta_1 \) and \( \theta_2 \).

In our example, dividing the magnitudes \( r_1 = 6 \) and \( r_2 = 7 \) results in \( \frac{6}{7} \). Subtracting the angles \( 40^{\circ} \) and \( 100^{\circ} \) gives us \(-60^{\circ} \). This is how the problem is reduced to a simpler form, \( \frac{6}{7}(\cos(-60^{\circ}) + i \sin(-60^{\circ})) \).
Understanding this property helps streamline complex numbers' division, transforming the operation into manageable steps.

Trigonometric Identities

Trigonometric identities allow us to simplify complex expressions by exploiting the properties of sine and cosine. This is particularly useful when handling angles that might seem cumbersome otherwise.

• The cosine function is an even function, meaning \( \cos(-x) = \cos(x) \).
• In contrast, sine is an odd function, implying \( \sin(-x) = -\sin(x) \).

In the given problem, these identities are applied as follows: \( \cos(-60^{\circ}) = \cos(60^{\circ}) \) and \( \sin(-60^{\circ}) = -\sin(60^{\circ}) \). These transformations help us represent the complex number without negative angles, allowing us to interpret values directly. Utilizing these identities ensures that each part of the expression is simplified correctly, facilitating the process of evaluating complex numbers in trigonometric form.

Simplifying Complex Numbers

Simplifying complex numbers involves reducing the expression to its most straightforward form while maintaining the integrity of its value. This process involves carefully handling both the real and imaginary components.

• After using the quotient property and trigonometric identities, we are left with the expression \( \frac{6}{7}(\cos(60^{\circ}) - i \sin(60^{\circ})) \).
• Values for \( \cos(60^{\circ}) \) and \( \sin(60^{\circ}) \) are common trigonometric values: \( \frac{1}{2} \) and \( \frac{\sqrt{3}}{2} \) respectively.

Therefore, substituting these into our expression gives us \( \frac{6}{7} \left( \frac{1}{2} - i \frac{\sqrt{3}}{2} \right) \).
Distributing \( \frac{6}{7} \) across both the real and imaginary parts, we get \( \frac{3}{7} - i \frac{3\sqrt{3}}{7} \).
This form is elegantly simplified and showcases the powerful application of trigonometric form handling and trigonometric properties in expressing complex numbers efficiently.