Problem 42 - 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. $$10\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right)$$

Step-by-Step Solution

Answer

The standard form of the given complex number is \(5\sqrt{5} + 5i\sqrt{5}\).

Step 1: Identify the magnitude and the argument of the complex number

Recognize that the given complex number is in the form of \(r(\cos{\theta}+i \sin{\theta})\), where \(r=10\) is the magnitude (radius) and \(\theta = \frac{2 \pi}{5}\) is the argument (angle formed with the positive x-axis).

Step 2: Convert from trigonometric form to standard form

Use the identities \(\cos{\theta} = \frac{e^{i\theta}+e^{-i\theta}}{2}\) and \(i \sin{\theta} = \frac{e^{i\theta}-e^{-i\theta}}{2i}\) to convert \(\cos{\theta}+i \sin{\theta}\) part into standard form. After substitution and simplification, one could obtain \(5\sqrt{5} + 5i\sqrt{5}\) which is the standard form for the given complex number.

Step 3: Verify using a graphing utility

To ensure the validity of the results, one could use a graphing utility to compare the given trigonometric form and the obtained standard form of the complex number. If they match, the result can be validated.

Key Concepts

Trigonometric Form

Complex numbers can be expressed in different forms, one of which is the trigonometric form. Often represented as \( r (\cos{\theta} + i \sin{\theta}) \), it neatly combines both magnitude \( r \) and angle \( \theta \) (or argument). This applies when the complex number is plotted on the polar coordinate system.
By using this form, you easily visualize the number's position with respect to distance from the origin (magnitude) and the angle it forms with the positive x-axis (argument).

When a complex number is given in this trigonometric form, it can be converted into other forms for further calculations or representations. Understanding this form is crucial as it provides insight into the geometric properties of complex numbers.

Standard Form

The standard form for a complex number is written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. Every complex number can be represented in this form, which is the most common and familiar to most people.

To convert from trigonometric form to standard form, one needs to calculate the values of \( \cos{\theta} \) and \( \sin{\theta} \), usually using known trigonometric identities or angles. For example, in the given exercise, you transform \( 10(\cos{ \frac{2 \pi}{5}} + i \sin{ \frac{2 \pi}{5}}) \) into \( 5\sqrt{5} + 5i\sqrt{5} \).
This conversion is practical when you need to perform arithmetic operations like addition and subtraction on complex numbers.

Magnitude and Argument

Magnitude and argument are two key components of a complex number when represented in trigonometric form. The magnitude, or modulus, is the distance of the complex number from the origin, often denoted as \( r \). The argument is the angle between the positive x-axis and the line representing the number, denoted as \( \theta \).

If you have a complex number \( z = r(\cos{\theta} + i\sin{\theta}) \), \( r = 10 \) and \( \theta = \frac{2\pi}{5} \) are directly observed from the expression given in the exercise. Calculating the exact values of these helps in converting and working with complex numbers in different forms.

Magnitude provides length while the argument offers direction, together making it simple to visualize and handle complex arithmetic.

Graphing Utility

A graphing utility, such as a graphing calculator or software, aids in verifying and visualizing complex numbers. By inputting the trigonometric form, you can confirm the calculations and conversions to standard form.

This tool is especially useful when you want to ensure that your transformations, like confirming the solution \( 5\sqrt{5} + 5i\sqrt{5} \), match the original problem. It serves as a reliable method for validation and helps avoid potential errors in manual calculations.
Graphing utilities can also display complex numbers graphically, enhancing understanding by providing a visual aspect of positions and relative differences. This visual approach can be especially helpful for learners who grasp better with graphical representations.