Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{5}+1=0$$

Step-by-Step Solution

Answer

The solutions are: \(x_k = \cos\left(\frac{\pi + 2k\pi}{5}\right) + i\sin\left(\frac{\pi + 2k\pi}{5}\right)\) for \(k = 0, 1, 2, 3, 4\).

Step 1: Rewrite in Standard Form

First, rewrite the equation in a standard form to solve for the complex roots. The equation given is \(x^5 + 1 = 0\). To make it suitable for solving, we write it as \(x^5 = -1\).

Step 2: Express "+1" as a Complex Number

Recognize that \(-1\) can be expressed using Euler's formula as a complex number: \(-1 = e^{i\pi}\). This representation helps in finding roots in trigonometric form, where \(\theta = \pi\) for the angle.

Step 3: Use the General Form for Complex Roots

The general form for the \(n\)-th roots of a complex number is given by: \(x = |r|^{1/n} \left[ \cos\left( \frac{\theta + 2k\pi}{n} \right) + i\sin\left( \frac{\theta + 2k\pi}{n} \right) \right]\). In our equation, \(|r| = 1\), \(\theta = \pi\), and \(n = 5\).

Step 4: Calculate the Fifth Roots

Calculate the fifth roots by substituting \(k = 0, 1, 2, 3, 4\) into the formula: \[x = \cos\left( \frac{\pi + 2k\pi}{5} \right) + i\sin\left( \frac{\pi + 2k\pi}{5} \right)\].

Step 5: List the Complex Solutions

Substitute the values of \(k\) into the formula from Step 4:- For \(k = 0\): \(x_1 = \cos\left(\frac{\pi}{5}\right) + i\sin\left(\frac{\pi}{5}\right)\)- For \(k = 1\): \(x_2 = \cos\left(\frac{3\pi}{5}\right) + i\sin\left(\frac{3\pi}{5}\right)\)- For \(k = 2\): \(x_3 = \cos\left(\pi\right) + i\sin\left(\pi\right)\)- For \(k = 3\): \(x_4 = \cos\left(\frac{7\pi}{5}\right) + i\sin\left(\frac{7\pi}{5}\right)\)- For \(k = 4\): \(x_5 = \cos\left(\frac{9\pi}{5}\right) + i\sin\left(\frac{9\pi}{5}\right)\).

Step 6: Present Solutions in Trigonometric Form

The complex solutions, expressed in trigonometric form, for \(x^5 + 1 = 0\) are: - \(x_1 = \cos\left(\frac{\pi}{5}\right) + i\sin\left(\frac{\pi}{5}\right)\)- \(x_2 = \cos\left(\frac{3\pi}{5}\right) + i\sin\left(\frac{3\pi}{5}\right)\)- \(x_3 = \cos(\pi) + i\sin(\pi)\)- \(x_4 = \cos\left(\frac{7\pi}{5}\right) + i\sin\left(\frac{7\pi}{5}\right)\)- \(x_5 = \cos\left(\frac{9\pi}{5}\right) + i\sin\left(\frac{9\pi}{5}\right)\).

Key Concepts

Trigonometric Form

Complex numbers can be represented in trigonometric form, also known as polar form, which provides a unique perspective on how to view these intriguing numbers. Rather than using the typical Cartesian coordinates (real and imaginary parts), we refer to the distance and angle from the origin in the complex plane. This distance is called the modulus, denoted as \(|z|\), and the angle is called the argument, represented as \( \theta \).
For a complex number \( z = a + bi \), its trigonometric form is given by:

• \( z = |z| (\cos(\theta) + i\sin(\theta)) \)

This form can simplify calculations, especially when dealing with powers and roots. It involves determining the modulus \(|z| = \sqrt{a^2 + b^2} \) and argument \( \theta = \tan^{-1}(\frac{b}{a}) \). In practice, this approach is particularly useful in problems involving Euler's formula and roots of complex numbers.

Euler's Formula

Euler's Formula is one of the most beautiful and profound results in mathematics. It connects complex exponentials and trigonometric functions in a simple yet powerful way, expressed as:

• \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \)

Using this formula, any complex number can be written as \( z = |z|e^{i\theta} \), combining both modulus and argument approaches.
Euler's formula is instrumental in rewriting complex numbers in exponential form, which is especially helpful when raising numbers to powers or finding roots. For example, in the equation \( x^5 + 1 = 0 \), the number \(-1\) is represented using Euler's formula as \( e^{i\pi} \), making it easier to find the fifth roots by expressing them in trigonometric form.
Euler's connection of exponential and trigonometric functions gives us a more profound understanding of the underlying structure of complex numbers and their behavior in different scenarios.

Nth Roots of Unity

Finding the nth roots of unity involves calculating the complex roots of the equation \( x^n = 1 \), where \( n \) is a positive integer. These roots form a remarkable pattern on the complex plane, evenly spaced on a unit circle.
The general form for finding the \( n \)-th roots of a complex number \( r \cdot e^{i\theta} \) is expressed as:

• \( x = |r|^{1/n} \left[ \cos\left( \frac{\theta + 2k\pi}{n} \right) + i\sin\left( \frac{\theta + 2k\pi}{n} \right) \right] \)

Where \( k = 0, 1, 2, \ldots, n-1 \). These roots are often referred to as the "roots of unity" when \( r = 1 \).
In the specific example of \( x^5 = -1 \), the fifth roots are determined by substituting \( r = 1 \) and \( \theta = \pi \), resulting in evenly spaced complex solutions around the unit circle. This symmetry and structured pattern can simplify the understanding of solutions and their applications in diverse fields like signal processing and quantum mechanics.