Problem 71 - Algebra and Trigonometry

Question

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fifth roots of 32

Step-by-Step Solution

Answer

The solution includes complex numbers in rectangular form that are the fifth roots of 32.

Step 1: Definition of a Complex Root

A complex nth root of a number x is a complex number z such that \(z^n = x\). In this case, it's requested to find the fifth roots of 32.

Step 2: Conversion to Polar Form

Before applying De Moivre’s theorem, the number 32 needs to be expressed in polar form. 32 is a real number, so in polar form it becomes \(32\(\cos(0) + i\sin(0)\) = 32\).

Step 3: Application of De Moivre’s Theorem

For each k from 0 to 4, find the root using the formula \(root_k = 32^{1/5}[cos\((2k\pi/5 + 2\pi m)/5\) + isin\((2k\pi/5 + 2\pi m)/5\)]\), where m is an integer. This formula results from De Moivre’s theorem.

Step 4: Calculation and Conversion into Rectangular Form

Each root_k obtained above should be rounded to the nearest tenth where necessary and then transformed back into rectangular form. This can be done by applying the Euler’s formula \(e^{i\theta} = cos\theta + isin\theta\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for finding powers and roots of complex numbers. It states that if you have a complex number in polar form, given by \(r(\cos(\theta) + i\sin(\theta))\), you can easily find its powers or roots. The theorem is applied as:

• For powers: \((r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta))\).
• For roots: \(r^{1/n}(\cos((\theta+2k\pi)/n) + i\sin((\theta+2k\pi)/n))\) where \(k = 0, 1, \ldots, n-1\).

De Moivre's Theorem simplifies the mathematical operations for complex numbers by transforming them into polar coordinates where adding/subtracting angles and multiplying/dividing magnitudes is straightforward. This is especially useful in finding roots of unity and solving polynomial equations.

Polar Form

The polar form of a complex number is an alternative way to represent it, using the magnitude and angle instead of the standard rectangular form. The rectangular form \(a + bi\) can be converted into polar form \(r(\cos \theta + i\sin \theta)\) where:

• \(r\) (the modulus) is the distance from the origin to the point \((a, b)\) in the complex plane, calculated as \(r = \sqrt{a^2 + b^2}\).
• \(\theta\) (the argument) is the angle with the positive x-axis, found using \(\theta = \tan^{-1}(\frac{b}{a})\).

In polar form, complex multiplication and division become simpler. You multiply or divide the magnitudes and add or subtract the angles. For instance, multiplying two complex numbers in polar form simply involves multiplying their magnitudes and adding their angles. Polar form is essential when using De Moivre's Theorem.

Rectangular Form

Rectangular form represents complex numbers in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. It's called "rectangular" because in a complex plane, these coordinates form a right triangle with \(a\) serving as the base and \(b\) as the height.
Converting between rectangular and polar forms is often necessary in complex number operations, as some equations are simpler in one form than the other. For example, to find complex roots using De Moivre's Theorem, the simplest way is often to first convert to polar form and then back to rectangular form after calculations.
In practice, after calculating angles and magnitudes to find roots with polar form, you convert back to \(a + bi\) to express the final answer for consistent interpretation, often required by contexts like definitive homework results.

Euler's Formula

Euler's Formula is a critically important concept that links complex numbers and exponential functions. Euler's formula states:
\[ e^{i\theta} = \cos \theta + i\sin \theta \]This is a powerful equation because it allows us to use the exponentiation of complex numbers with ease and demonstrates the deep relationship between trigonometry and complex numbers. Using Euler’s formula, a complex number can be expressed as \(re^{i\theta}\), especially beneficial when working with De Moivre's Theorem.
Euler's formula simplifies conversion between polar and rectangular forms. It provides a way to work with angles and engineering problems more intuitively by applying exponential functions. The ability to handle complex exponentials is essential in fields that deal with waves and oscillations, such as electrical engineering and quantum mechanics.