Problem 77 - Precalculus with Limits

Question

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([5(\cos\ 3.2 + i\ \sin\ 3.2)]^{4}\)

Step-by-Step Solution

Answer

The 4th power of the complex number in standard form is approximately \(369.572 + 445.202i\).

Step 1: Identify the Modulus and Argument of the Complex Number

In the complex number \(5(\cos\ 3.2 + i\ \sin\ 3.2)\), the modulus is 5 and the argument is 3.2.

Step 2: Apply DeMoivre's Theorem

DeMoivre's Theorem states that \((r(cos\ θ + i\ sin\ θ))^n = r^n((cos(nθ) + i\ sin(nθ)). According to this rule: \([5(\cos\ 3.2 + i\ \sin\ 3.2)]^{4} = 5^4(\cos(4*3.2) + i\ \sin(4*3.2))\). Upon calculating, the result is \(625(\cos\ 12.8 + i\ \sin\ 12.8)\).

Step 3: Convert the Result to Standard Form

The standard form of a complex number is a + ib, where a is the real part and b is the imaginary part. In order to convert the polar form to standard form, we use the identities \(\cos\ 12.8 = \cos\ (2\pi*2 + 0.8) = \cos\ 0.8\) and \(\sin\ 12.8 = \sin\ (2\pi*2 + 0.8) = \sin\ 0.8\). Therefore, we calculate the standard form as \(625*\cos\ 0.8 + 625i*\sin\ 0.8\).

Key Concepts

Complex Numbers

Complex numbers are fundamental in mathematics, especially when dealing with equations involving square roots of negative numbers. A complex number is expressed in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\).
Complex numbers can represent two-dimensional quantities and can be visualized on the complex plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part.

• Real Part (a): The value along the horizontal axis.
• Imaginary Part (b): The value along the vertical axis, represented by \(bi\).

By using complex numbers, we can easily perform mathematical operations that account for two-dimensional space instead of just one-dimensional real numbers. This is particularly useful in various fields of science and engineering, where multi-dimensional modeling is crucial.

Modulus and Argument

The modulus and argument help us understand the size and direction of a complex number when visualized in the complex plane. The modulus, also known as the magnitude, is the distance of the complex number from the origin on the complex plane. For a complex number \(a + bi\), the modulus is calculated as \(\sqrt{a^2 + b^2}\). In the exercise given, the modulus is \(5\), meaning the complex number is 5 units away from the origin.
The argument is the angle formed with the positive real axis. It's usually denoted by \(\theta\) and measured in radians or degrees. It represents the direction of the complex number. The argument in the example is given as \(3.2\) radians.

• Modulus (r): Can be thought of as the "length" of the complex number.
• Argument (θ): The angle with respect to the positive x-axis.

Understanding these two components is key to manipulating complex numbers, especially when they are in polar form.

Polar to Standard Form Conversion

Converting from polar to standard form is an essential process to express complex numbers in a more familiar \(a + ib\) form.
Polar Form of a complex number is given by \(r(\cos \theta + i\sin \theta)\), where \(r\) is the modulus and \(\theta\) is the argument. This form is very handy, especially when multiplying or raising complex numbers to a power, as seen in DeMoivre's Theorem.
Standard Form, also known as the rectangular form, is expressed as \(a + bi\). To convert:

• Calculate \(a = r \cos \theta\)
• Calculate \(b = r \sin \theta\)

For example, after applying DeMoivre's Theorem, we have \(625(\cos 12.8 + i\sin 12.8)\). By evaluating \(\cos 12.8\) and \(\sin 12.8\), and multiplying them by \(625\), we achieve the equivalent compound number in standard form.
This conversion allows us to work in either form depending on what is most efficient for the calculations we need to perform.