Problem 41 - Precalculus with Limits

Question

Writing a Complex Number in Standard Form In Exercises \(41-44,\) use a graphing utility to write the complex number in standard form. $$5\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right)$$

Step-by-Step Solution

Answer

The complex number \(5(\cos(\frac{\pi}{9}) + i\sin(\frac{\pi}{9}))\) in standard form is approximately \(4.879 + 1.736i\).

Step 1: Note the Formulas

It should be noted that the complex number given is in the form of \(re^{i\Theta}\) where \(r = 5\) is the magnitude (the distance from the origin to the dot in the complex plane) and \(\Theta = \frac{\pi}{9}\) is the angle. The task is to convert it into the standard form, which is \(a + bi\). For this, we can use Euler's formula, \(r(\cos(\Theta) + i\sin(\Theta))\), which represents the same complex number.

Step 2: Apply Euler's Formula to Convert Form

We apply the formula using the given value of magnitude and angle: \(r\cos(\Theta) + ri\sin(\Theta)\), which equals \(5\cos(\frac{\pi}{9}) + 5i\sin(\frac{\pi}{9})\). We'll calculate these values independently.

Step 3: Calculate the Real Part (a)

The real part of the number will be \(a = r\cos(\Theta) = 5\cos(\frac{\pi}{9})\). Using a calculator, this comes out to approximately 4.879.

Step 4: Calculate the Imaginary Part (b)

The imaginary part of our number will be \(b = ri\sin(\Theta) = 5i\sin(\frac{\pi}{9}\). Using a calculator, this is approximately 1.736i.

Step 5: Write the Complex Number in Standard Form

Combine the real and imaginary parts to write the complex number in standard form, which is in the form \(a + bi\). So, here we have it as approx. \(4.879 + 1.736i\).

Key Concepts

Euler's Formula

Euler's formula is a beautiful relationship in mathematics that connects complex exponentials with trigonometric functions. It is written as \( e^{i\Theta} = \cos(\Theta) + i\sin(\Theta) \).
In the context of complex numbers, this formula allows us to express a complex number in exponential form \( re^{i\Theta} \), where \( r \) is the magnitude and \( \Theta \) is the angle.
This is incredibly useful because it simplifies the process of manipulating complex numbers, especially when dealing with multiplication and division. It turns complex multiplication into addition of angles, which is often easier to handle.

• You can think of Euler's formula as a bridge between the polar and rectangular coordinates in the complex plane.
• It helps visualize complex numbers on a unit circle where conventional trigonometric identities come into play.
• In practical applications, Euler's formula is foundational in fields like electrical engineering, quantum mechanics, and signal processing.

Standard Form

The standard form of a complex number is \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This form is the most common way to express complex numbers.
Converting complex numbers from other forms to the standard form is an essential skill. It allows easier arithmetic operations like addition and subtraction.
To convert a polar or exponential form to standard form, you can use trigonometric functions to find \( a \) and \( b \). For example, given \( re^{i\Theta} \), calculate:

• The real part: \( a = r\cos(\Theta) \)
• The imaginary part: \( b = r\sin(\Theta) \)

Together, these give you the standard form \( a + bi \). This explicit representation makes it much easier to understand and work with complex numbers in various mathematical contexts.

Graphing Utility

A graphing utility is a tool, such as a graphing calculator or computer software, used to visually represent mathematical concepts. When working with complex numbers, graphing utilities can be particularly helpful to:

• Visualize the location of a complex number in the complex plane.
• Convert between forms like polar to standard, as shown by plotting the magnitude and angle directly.
• Conduct complex operations such as addition, subtraction, multiplication, and division visually, aiding better understanding.

Using a graphing utility often helps in verifying calculations and intuitively understanding the effects of operations on complex numbers. For instance, in our exercise, a graphing utility would confirm the conversion of a polar form complex number to standard form by displaying it clearly on the complex plane.
It is beneficial for learners in reinforcing concepts by offering a visual dimension to theoretical calculations.

Magnitude and Angle Conversion

Magnitude and angle conversion form the basis for expressing complex numbers in different forms. The magnitude of a complex number is essentially its distance from the origin in the complex plane, calculated as \( r = \sqrt{a^2 + b^2} \).
The angle, or argument, gives us its direction, calculated using trigonometric ratios such as \( \Theta = \tan^{-1}(b/a) \).
This conversion is crucial when switching from polar to standard form or vice versa. The magnitude and angle determine the exponential or polar form \( re^{i\Theta} \), which can then be converted to the standard form using:

• The real part: \( a = r\cos(\Theta) \)
• The imaginary part: \( b = r\sin(\Theta) \)

By understanding how to calculate and convert between these forms, you can solve complex number problems more efficiently. This knowledge is essential for applications across physics, engineering, and other technical fields, where visualizing and manipulating complex numbers is often required.