Problem 91 - Algebra and Trigonometry

Question

Solve the equation. $$ z^{3}+1=-i $$

Step-by-Step Solution

Answer

The solutions are \(z = \sqrt[6]{2} \text{cis}(-\pi/4)\), \(z = \sqrt[6]{2} \text{cis}(5\pi/12)\), and \(z = \sqrt[6]{2} \text{cis}(-7\pi/12)\).

Step 1: Rearrange the Equation

The given equation is \(z^3 + 1 = -i\). To solve for \(z\), first isolate \(z^3\) by rewriting the equation as \(z^3 = -i - 1\).

Step 2: Simplify the Right Side

Combine like terms on the right side: \(-i - 1 = -1 - i\). Our equation now is \(z^3 = -1 - i\).

Step 3: Express the Complex Number in Polar Form

Now, express \(-1 - i\) in polar form: the modulus \(r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}\) and the argument \(\theta = \text{atan2}(-1, -1) = -\frac{3\pi}{4}\). Thus, \(-1 - i = \sqrt{2}\left(\cos\left(-\frac{3\pi}{4}\right) + i\sin\left(-\frac{3\pi}{4}\right)\right)\).

Step 4: Apply De Moivre's Theorem

To find \(z\), use the fact that \(z = (r \text{cis} \theta)^{1/3}\). This gives: \(z = (\sqrt{2})^{1/3} \text{cis}\left(\frac{-3\pi}{12} + \frac{2k\pi}{3}\right)\) for \(k = 0, 1, 2\).

Step 5: Calculate the Roots

Calculate the three cube roots: 1. For \(k = 0\), \(z_0 = \sqrt[6]{2} \text{cis}\left(-\frac{\pi}{4}\right)\). 2. For \(k = 1\), \(z_1 = \sqrt[6]{2} \text{cis}\left(-\frac{\pi}{4} + \frac{2\pi}{3}\right)\). 3. For \(k = 2\), \(z_2 = \sqrt[6]{2} \text{cis}\left(-\frac{\pi}{4} + \frac{4\pi}{3}\right)\).

Step 6: Convert Roots Back to Cartesian Form (optional)

Convert the complex solutions from polar form back to Cartesian form if needed: 1. \(z_0 = \sqrt[6]{2} \left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\). 2. \(z_1 = \sqrt[6]{2} \left(\cos\left(\frac{5\pi}{12}\right) + i\sin\left(\frac{5\pi}{12}\right)\right)\). 3. \(z_2 = \sqrt[6]{2} \left(\cos\left(-\frac{7\pi}{12}\right) + i\sin\left(-\frac{7\pi}{12}\right)\right)\). The exact values might be computed based on the context and requirements.

Key Concepts

Polar Form

Polar form is a way of representing complex numbers using a combination of a modulus and an angle. The modulus, also known as the magnitude, is the distance from the origin to the complex number in the complex plane. The angle, usually denoted as \( \theta \), is measured in radians and is the angle formed with the positive real axis.Converting a complex number from Cartesian form \((a + bi)\) to polar form involves calculating:

• Modulus \(r\): \( r = \sqrt{a^2 + b^2} \)
• Argument \(\theta\): \( \theta = \text{atan2}(b, a) \)

Thus, a complex number \(a + bi\) in polar form becomes \(r(\cos \theta + i \sin \theta)\), often abbreviated as \(r \text{cis} \theta\). This form is particularly useful for multiplication and division of complex numbers as it simplifies many operations.

De Moivre's Theorem

De Moivre’s Theorem is a powerful formula for raising complex numbers in polar form to a power. It states that for any real number \(n\) and a complex number in polar form \(r(\cos \theta + i \sin \theta)\), \[(z)^n = r^n \left(\cos(n\theta) + i \sin(n\theta)\right)\]This theorem simplifies the computation by transforming the power operation into simple multiplication of the angle. For roots, we adjust the angle by \(\frac{2k\pi}{n}\) for each possible solution, where \(k\) is an integer. For example, to find cube roots, we compute the third roots of the modulus and divide the angle by three while offsetting it by multiples of \(\frac{2\pi}{3}\).De Moivre’s Theorem is essential when solving equations involving powers of complex numbers, allowing straightforward calculation of all possible solutions.

Complex Numbers

Complex numbers extend the idea of the number line to a plane using the imaginary unit \(i\), where \(i^2 = -1\). This creates numbers in the form \(a + bi\) where \(a\) and \(b\) are real numbers.Complex numbers can represent a wide variety of mathematical phenomena not easily represented on the real number line. They are particularly useful in engineering, physics, and applied mathematics for solving real-world problems that involve two-dimensional quantities.

• Real Part: The component \(a\) in \(a + bi\)
• Imaginary Part: The component \(b\) in \(a + bi\)

These numbers allow for the representation of cyclic behaviors, growth, and decay in systems where limits are otherwise required.

Cartesian Form

Cartesian form of complex numbers refers to expressing them as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Named after René Descartes, this form is instrumental in standard mathematical and scientific contexts for describing and manipulating complex numbers.To convert a complex number from polar back to Cartesian form, one uses:

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

This form is particularly useful for addition and subtraction of complex numbers, as these operations are performed directly on the real and imaginary components separately.For example, if a complex number is given in polar form as \(r \text{cis} \theta\), converting it back to Cartesian form helps in performing operations like comparing or adding to other complex numbers in the same form. The utility of shifting between polar and Cartesian forms allows complex mathematical analyses to become simpler and more intuitive.