Problem 52 - Algebra and Trigonometry

Question

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\sqrt{3} e^{i \frac{5 \pi}{18}}\right]^{6} $$

Step-by-Step Solution

Answer

Rectangular Form: \( \frac{27}{2} - \frac{27\sqrt{3}}{2} i \), Exponential Form: \( 27 e^{i \frac{5 \pi}{3}} \)

Step 1: Simplify the exponent

First, simplify the exponent of the given expression. If the expression is \(\big[\sqrt{3} e^{i \frac{5 \pi}{18}}\big]^{6}\), we use the exponent rule \( \left(a^m\right)^n = a^{mn} \). Here, the exponent becomes \((\sqrt{3})^{6} e^{i \frac{5 \pi}{18} \cdot 6}\).

Step 2: Calculate each part

Calculate the simplified parts separately. Since \( (\sqrt{3})^{6} = 3^{3} = 27\) and \( i \frac{5 \pi}{18} \cdot 6 = i \frac{5 \pi}{3} \), the expression simplifies to \( 27 e^{i \frac{5 \pi}{3}} \).

Step 3: Convert to rectangular form

To convert to rectangular form \(x + yi\), use the Euler's formula \( r e^{i \theta} = r (\cos \theta + i \sin \theta) \). Here, \( r = 27 \) and \( \theta = \frac{5 \pi}{3} \). Calculate \( \cos \frac{5 \pi}{3} \) and \( \sin \frac{5 \pi}{3} \): \( \cos \frac{5 \pi}{3} = \frac{1}{2} \) and \( \sin \frac{5 \pi}{3} = -\frac{\sqrt{3}}{2} \). Therefore, \( 27 ( \cos \frac{5 \pi}{3} + i \sin \frac{5 \pi}{3}) = 27 \bigg(\frac{1}{2} - i \frac{\sqrt{3}}{2}\bigg) = \frac{27}{2} - 27 i \frac{\sqrt{3}}{2}\). This simplifies to \( \frac{27}{2} - \frac{27\sqrt{3}}{2}i \).

Step 4: Write the final rectangular form

The rectangular form of the given expression is \( \frac{27}{2} - \frac{27\sqrt{3}}{2} i \).

Step 5: Verify the exponential form

Reconfirm the exponential form is \( 27 e^{i \frac{5 \pi}{3}} \).

Key Concepts

rectangular form

In mathematics, particularly in complex numbers, the rectangular form (or algebraic form) is a method of representing complex numbers. A complex number is written as:
\[ z = x + yi \]
where:

• \( x \) is the real part
• \( y \) is the imaginary part
• \( i \) is the imaginary unit, satisfying \( i^2 = -1 \)

For example, the complex number \( \frac{27}{2} - \frac{27\text{√3}}{2}i \) is in rectangular form, where:

• Real part \( x = \frac{27}{2} \)
• Imaginary part \( y = -\frac{27\text{√3}}{2} \)

Rectangular form is very useful for addition and subtraction of complex numbers.

exponential form

The exponential form of complex numbers is another way to express complex numbers, making use of the polar coordinate system. It's written as:
\[ z = re^{i\theta} \]
Here:

• \( r \) is the magnitude (or modulus) of the complex number
• \( \theta \) is the argument (or angle)

In our exercise, the complex number was found in exponential form as \( 27e^{i \frac{5π}{3}} \). Here, \( r = 27 \) and \( \theta = \frac{5π}{3} \). This form is especially helpful for multiplication and division of complex numbers due to the properties of exponents and the Euler's formula.

Euler's formula

Euler's formula is a fundamental bridge between the exponents and trigonometry, defined as:
\[ e^{i\theta} = \text{cos} \theta + i \text{sin} \theta \]
This formula allows us to convert between exponential form and rectangular form easily. For example, in the problem:
\[ 27e^{i \frac{5π}{3}} \]
Using Euler's formula, we can write:
\[ 27 (\text{cos} \frac{5π}{3} + i\text{sin} \frac{5π}{3}) \]
Where:

• \( \text{cos} \frac{5π}{3} = \frac{1}{2} \)
• \( \text{sin} \frac{5π}{3} = -\frac{\text{√3}}{2} \)

Thus, we can find the rectangular form from exponential form smoothly.

trigonometric functions

Trigonometric functions, such as sine and cosine, are vital in the study of complex numbers, especially when converting between different forms. In Euler's formula:
\[ e^{i\theta} = \text{cos} \theta + i \text{sin} \theta \]
both sine and cosine functions are used. These functions help describe the position of the complex number in the complex plane.
For example, to convert \( 27e^{i \frac{5π}{3}} \) to rectangular form, we used the trigonometric values:

• \( \text{cos} \frac{5π}{3} = \frac{1}{2} \)
• \( \text{sin} \frac{5π}{3} = -\frac{\text{√3}}{2} \)

This yielded the rectangular form:
\[ 27 \bigg(\frac{1}{2} - i \frac{\text{√3}}{2}\bigg) = \frac{27}{2} - 27 \frac{\text{√3}}{2}i \]