Problem 69 - 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 fourth roots of \(81\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)\)

Step-by-Step Solution

Answer

The complex fourth roots in rectangular form are \(\frac{3}{2} + i\frac{3\sqrt{3}}{2}\), \(-\frac{3\sqrt{3}}{2} + i\frac{3}{2}\), \(0 - 3i\), and \(\frac{3\sqrt{3}}{2} - i\frac{3}{2}\)

Step 1: Identify the square root

The complex number is \(81\left(\cos \frac{4 \pi}{3}+i \sin \frac{4\pi}{3}\right)\). First, look for the fourth roots of the number. Remember in polar form, we simply divide the angle by the root and find the fourth root of the magnitude. Here the magnitude is 81 and the angle is \(\frac{4 \pi}{3}\).

Step 2: Calculate the roots

The fourth root of 81 is 3; so, the magnitude of the roots is 3. The angles of the roots are \(\frac{\frac{4 \pi}{3}}{4}, \frac{\frac{4 \pi}{3}+2\pi}{4}, \frac{\frac{4 \pi}{3}+4\pi}{4}, \frac{\frac{4 \pi}{3}+6\pi}{4}\) which simplifies to \(\frac{\pi}{3}, \frac{5 \pi}{6}, \frac{3 \pi}{2}, \frac{13 \pi}{6}\). Thus the roots in polar form are \(3\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\), \(3\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)\), \(3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)\) and \(3\left(\cos \frac{13 \pi}{6}+i \sin \frac{13 \pi}{6}\right)\)

Step 3: Convert to rectangular form

Convert each of roots to rectangular form as follows: \(3\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) = 3(\frac{1}{2} + i\frac{\sqrt{3}}{2})\) (simplifies to \(\frac{3}{2} + i\frac{3\sqrt{3}}{2}\)), \(3\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right) = 3(-\frac{\sqrt{3}}{2} + i\frac{1}{2})\) (simplifies to \(-\frac{3\sqrt{3}}{2} + i\frac{3}{2}\)), \(3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right) = 3(0 - 1i)\) (simplifies to \(0 - 3i\)), and \(3\left(\cos \frac{13 \pi}{6}+i \sin \frac{13 \pi}{6}\right) = 3(\frac{\sqrt{3}}{2} - i\frac{1}{2})\) (simplifies to \(\frac{3\sqrt{3}}{2} - i\frac{3}{2}\)).

Key Concepts

Rectangular Form

In mathematics, complex numbers are often expressed in two main forms: rectangular and polar. Rectangular form is the most common way to represent complex numbers. It expresses a complex number as a combination of a real part and an imaginary part.
A complex number in rectangular form looks like this:
\(a + bi\)

• "\(a\)" is the real part.
• "\(b\)" is the imaginary part, where \(i\) is the imaginary unit, defined as \(i^2 = -1\).

The beauty of the rectangular form is its simplicity in addition and subtraction operations. It aligns closely with Cartesian coordinates in the complex plane, which you might visualize as a graph where each point corresponds to a complex number. This makes it easy to visualize operations like addition as geometric translations. For the conversion from the polar form used earlier, you can use trigonometric identities:

• Real part \(= r \cos \theta\)
• Imaginary part \(= r \sin \theta\)

Where \(r\) is the magnitude and \(\theta\) is the angle in polar form. Making these conversions between rectangular and polar forms is crucial when solving complex number equations.

Polar Form

Polar form provides a different and sometimes more intuitive way to express complex numbers, especially when they are involved in multiplication, division, and finding roots.
A complex number in polar form is denoted by:
\(r(\cos \theta + i \sin \theta)\) or sometimes using Euler's formula \(re^{i\theta}\).

• "\(r\)" denotes the magnitude of the complex number.
• "\(\theta\)" is the argument, representing the angle in radians.

These values can be understood geometrically:

• Magnitude \(r\) is the distance from the origin to the point in the complex plane.
• Angle \(\theta\) describes the direction of the point from the real axis.

This form is especially useful when performing operations like multiplication and division, as it converts these operations into arithmetic operations on the magnitudes and angles. In this exercise, you converted from polar to rectangular form by calculating the real and imaginary parts. Polar form is powerful in scenarios like finding complex roots, where you can simply divide the angle by the root index and take the nth root of the magnitude.

Fourth Roots

When working with complex numbers, calculating roots is a unique process compared to real numbers. The exercise involved finding the fourth roots of a complex number. Here, a fourth root is any number that, when multiplied by itself four times, results in the original number.
For a complex number given in polar form, such as \(r(\cos \theta + i \sin \theta)\), the method involves:

• Taking the fourth root of the magnitude \(r\).
• Dividing the angle \(\theta\) by 4, since we're finding the fourth root.

Remember, according to De Moivre's Theorem, if a complex number has an angle \(\theta\), it also has other roots obtained by adding multiples of \(2\pi\) to \(\theta\) and then dividing by the root number (here it is 4).

• This ensures each distinct root fills an equal part of a circle in the complex plane.
• In this problem, we distributed angles evenly, resulting in distinctive fourth roots with each having a unique \(\theta_k = \frac{4\pi}{3} + \frac{2k\pi}{4}\) for \(k = 0, 1, 2, 3\).

Complex Roots

Understanding complex roots involves recognizing that every non-zero complex number has not just one nth root, but several, scattered evenly around a circle in the complex plane. Solving "complex roots" means finding all possible values of a root of a given complex number.
For instance, in our exercise, we needed to find fourth roots, meaning we were searching for four values that could return the original complex number when each is raised to the fourth power.

• Each complex root has a different angle but shares the same magnitude.
• These roots can be found using De Moivre's Theorem.

Mathematically, if \(z\) is a complex number, then its nth roots are generally expressed as:\[z_k = r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i \sin(\frac{\theta + 2k\pi}{n}))\]for integers \(k = 0, 1, 2,..., n-1\).

• This exercise's roots were derived using this approach, producing four distinct roots based upon distinct angles but each sharing the same reduced magnitude from the original complex number's magnitude.

This concept allows us to exploit the full symmetry of the complex plane and provides insights into phenomena that occur in higher-dimensional spaces.