Problem 83 - Precalculus with Limits

Question

Finding the \(n\) th Roots of a Complex Number \(\operatorname{In}\) Exercises \(81-96,(\) a) use the formula on page 446 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Cube roots of $$8\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The cube roots of the given complex number are obtained using De Moivre's theorem. These roots can be represented graphically on the complex plane with each root being a distance of the cube root of 8 from the origin at angles of \( \frac{\theta + 2k\pi}{3} \) for \(k = 0, 1, 2\). Each root can also be expressed in standard form as \( a + bi \).

Step 1: Calculate the modulus and argument of the complex number

The modulus \(r\) of the complex number \(8(\cos(\frac{2 \pi}{3}) + i \sin(\frac{2 \pi}{3}))\) is 8. The argument \(\theta\) is \(\frac{2 \pi}{3}\).

Step 2: Apply De Moivre's theorem to find the cube roots

Use the formula for the cube root \( r^{1/3} [\cos((\theta + 2k\pi) / 3) + i \sin ((\theta + 2k\pi) / 3)] \), where \(k = 0, 1, 2\). Substituting \( r = 8 \) and \( \theta = \frac{2\pi}{3} \) gives the three roots.

Step 3: Represent the roots graphically

Plot the roots on the complex plane. The roots are at an angle of \( \frac{\theta + 2k\pi}{3}\) from the positive x-axis, and a distance of \(8^{1/3}\) from the origin.

Step 4: Express the roots in standard form

Use the formula \( r(\cos \theta + i \sin \theta) = r[\cos(\theta) + i\sin(\theta)] = r \cdot cis(\theta) = a + bi \), where \(a = r\cos \theta\) and \(b = r\sin \theta \), to convert each of the cube roots to the standard form \(a + bi\).

Key Concepts

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For a complex number, there are actually three distinct cube roots. If you think about multiplying in three dimensions, it makes sense that a complex number's cube can lead back to the same point in three different ways.
The general formula for finding the cube roots of a complex number in polar form is \[ r^{1/3} \left[ \cos\left( \frac{\theta + 2k\pi}{3} \right) + i \sin\left( \frac{\theta + 2k\pi}{3} \right) \right] \]where:

• \( r \) is the modulus of the original complex number.
• \( \theta \) is the argument of the original complex number.
• \( k \) is an integer that can be 0, 1, or 2 to give the three different roots.

Each value of \( k \) gives a different cube root, all equidistant from each other in the complex plane. This symmetry is a beautiful aspect of complex numbers.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for simplifying powers and roots of complex numbers. The theorem states that for any complex number in polar form,\( r(\cos \theta + i \sin \theta) \), and any integer \( n \), it is represented as:\[ r^n (\cos(n\theta) + i \sin(n\theta)) \].
This formula allows us to raise complex numbers to any power, or extract roots just like you saw with cube roots.
To use this theorem for cube roots, we adjust \( n \) to the reciprocal of the cube, \( 1/3 \). Each \( k \) value results in a distinct cube root, spaced evenly around the circle. This is particularly useful when solving for roots as it transforms complex multiplication into a simpler process.
By using De Moivre's theorem, you can transform any complex power operation into a manageable step-by-step process, simplifying calculations and exponents.

Modulus and Argument

Every complex number can be described in terms of its modulus and argument. The modulus, \( r \), is the distance from the origin on the complex plane to the number itself. It can be visualized as the length of the vector pointing to the complex number.
The argument, \( \theta \), is the angle formed with the positive x-axis, indicating the direction from the origin.
For the complex number \( 8(\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3}) \), the modulus is 8, indicating it is 8 units away from the origin. The argument is \( \frac{2 \pi}{3} \), which helps in identifying its exact position over a circle.
These components make it easier to express complex numbers in polar form, which significantly simplifies the process of finding roots and powers using De Moivre's theorem.

Complex Plane Representation

The complex plane is a two-dimensional plane where every complex number corresponds to a point. This plane is similar to the Cartesian plane, but it uses the x-axis for real parts and the y-axis for imaginary parts.
When representing cube roots on the complex plane, the roots are plotted as points equidistant from each other around a circle centered at the origin. The circle’s radius is the cube root of the modulus of the original complex number.
For example, the cube roots of \( 8(\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3}) \) are positioned at a distance of \( 8^{1/3} \) from the origin. Since we have three distinct cube roots, they are spread out evenly on the circle, each separated by the same angle.
The graphical representation is useful as it visually demonstrates the symmetry and spacing of the roots, helping to strengthen the understanding of complex roots.