Question

In Problems \(45-56,\) write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[4\left(\cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9}\right)\right]^{3} $$

Short answer

Rectangular form: \[-32 + 32i\sqrt{3} \]. Exponential form: \[ 64 e^{i\frac{2\pi}{3}} \]

Step-by-step solution

- Identify the given expression

Start with the given expression: \ \[4\left(\cos \frac{2 \pi}{9} + i \sin \frac{2 \pi}{9}\right)\right]^3 \]

- Recognize the form

Notice that the given expression is in polar form. Recognize that \[ 4 \left(\cos \frac{2 \pi}{9} + i \sin \frac{2 \pi}{9} \right) \] can be expressed as \( 4 e^{i\frac{2\pi}{9}} \) using Euler's formula.

- Apply De Moivre's Theorem

Apply De Moivre's Theorem to the given expression. This theorem states that \[ \left(r e^{i\theta}\right)^n = r^n e^{in\theta} \]. So, \[ \left[4\left(\cos\frac{2\pi}{9} + i \sin\frac{2\pi}{9}\right)\right]^3 = \left(4 e^{i\frac{2\pi}{9}}\right)^3 = 4^3 e^{i 3\left(\frac{2\pi}{9}\right)} \]

- Simplify the expression

Simplify the result from the previous step: \[ 4^3 e^{i \left(\frac{6\pi}{9}\right)} = 64 e^{i \frac{2\pi}{3}} \]

- Exponential form

The simplified result in exponential form is \[ 64 e^{i\frac{2\pi}{3}} \]

- Calculate rectangular form

To convert to rectangular form, use Euler's formula again: \[ 64 e^{i\frac{2\pi}{3}} = 64 (\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) \]. Then, compute the cosine and sine: \[ \cos \frac{2\pi}{3} = -\frac{1}{2} \] \[ \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \] Therefore, \[ 64 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = -32 + 32i\sqrt{3} \]

- Rectangular form

The simplified result in rectangular form is \[-32 + 32i\sqrt{3} \]

Key concepts

Rectangular Form

Rectangular form, also known as Cartesian form, represents a complex number as a combination of a real part and an imaginary part. It looks like this: \( x + yi \), where \( x \) is the real part and \( y \) is the imaginary part.
To understand how to convert from other forms to rectangular form, it's important to grasp these basics:

• Real and Imaginary Parts: The real part, \( x \), is a regular number, while the imaginary part, \( y \), multiplies by \( i \) (the square root of -1).
• Addition and Subtraction: We combine real parts with real parts and imaginary parts with imaginary parts.
• Example Conversion: If we convert an exponential form\( 64 e^{i \frac{2\pi}{3}} \) into rectangular form, we use Euler's formula (\( e^{i\theta} = \cos\theta + i\sin\theta \)).
  
  So,\( 64 e^{i \frac{2\pi}{3}} \) turns into \( 64 (\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) \).
  Next, find the values of the trigonometric functions: \( \cos \frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \).
  Multiply these by 64, getting:\( 64 (-\frac{1}{2} + i \frac{\sqrt{3}}{2}) = -32 + 32i\sqrt{3} \)
  This final result is the rectangular form: \( -32 + 32i\sqrt{3} \).

Understanding how to break down a complex number into real and imaginary parts makes working with them far simpler.

Exponential Form

The exponential form of a complex number leverages Euler's formula to express complex numbers compactly. This form is defined as \( re^{i\theta} \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle).
Here's what you need to know:

• Magnitude \( r \): This is the distance from the origin to the point in the complex plane, calculated as \( \sqrt{x^2 + y^2} \).
• Argument \( \theta \): This is the angle formed with the positive real axis, found using \( \tan^{-1}(\frac{y}{x}) \).
• Using Euler's Formula: Euler's formula states that \( e^{i\theta} = \cos\theta + i\sin\theta \). This helps us switch between forms efficiently.
  
  For example, consider the expression \( 64 e^{i\frac{2\pi}{3}} \):
  To find the exponential form, apply De Moivre's Theorem: \( \left[4 e^{i\frac{2\pi}{9}}\right]^3 = 4^3 e^{i(3 \cdot \frac{2\pi}{9})} = 64 e^{i\frac{2\pi}{3}} \).
• Simplifying: The simplified exponential form is \( 64 e^{i\frac{2\pi}{3}} \). This concise representation is particularly useful for complex computations involving powers and roots.

Understanding exponential form helps simplify multiplication, division, and exponentiation of complex numbers.

Euler's Formula

Euler's formula provides a deep connection between trigonometric functions and the exponential function. It's written as \( e^{i\theta} = \cos\theta + i\sin\theta \) and has profound implications in mathematics, particularly in complex number theory.
Key concepts include:

• Bridge Between Forms: Euler's formula allows us to convert between exponential and rectangular forms seamlessly. This conversion is essential for solving many problems.
• Application: Using the formula, an expression like \( 64 e^{i\frac{2\pi}{3}} \) can be transformed into rectangular form by expanding \( e^{i\theta} \) as \( \cos\theta + i\sin\theta \).
  For instance, \( 64 e^{i\frac{2\pi}{3}} \) converts to \( 64 (\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}) \). This is \( 64(-\frac{1}{2} + i\frac{\sqrt{3}}{2}) \), which simplifies to \( -32 + 32i\sqrt{3} \).
• Simplifying Calculations: Euler's formula makes dealing with powers and roots more manageable. For instance, raising a complex number in polar form to a power is straightforward using De Moivre's Theorem.
  Given \( (re^{i\theta})^n = r^n e^{in\theta} \), calculate: \( \left[4e^{i\frac{2\pi}{9}}\right]^3 = 64 e^{i\frac{2\pi}{3}} \).

Euler's formula is a powerful tool, unlocking simpler methods to handle complex number operations and making transitions between different forms of expression much easier.