Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Find the four fourth roots of \(-8+8 \sqrt{3} i\).

Step-by-Step Solution

Answer

The four fourth roots are 2 cis(\(\frac{5\pi}{24}\)), 2 cis(\(\frac{11\pi}{24}\)), 2 cis(\(\frac{17\pi}{24}\)), and 2 cis(\(\frac{23\pi}{24}\)).

Step 1: Express the Complex Number in Polar Form

To simplify finding the roots, we first express the complex number in polar form. The complex number is \(-8 + 8\sqrt{3}i\). Calculate the modulus: \\[r = \sqrt{(-8)^2 + (8\sqrt{3})^2} = \sqrt{64 + 192} = \sqrt{256} = 16\]\Now find the argument, \(\theta\), using the tangent function. Since tan(\(\theta\)) = \(\frac{8\sqrt{3}}{-8}\) = -\(\sqrt{3}\), the argument \(\theta\) is in the second quadrant (as the complex number is in the second quadrant): \[\theta = 150^\circ = \frac{5\pi}{6}\]\Thus, the polar form is \(16 \text{{cis}} \frac{5\pi}{6}\).

Step 2: Calculate the Fourth Roots Using De Moivre's Theorem

De Moivre's Theorem states that the nth root of a complex number is given by \[r^{1/n} \text{{cis}} \left(\frac{\theta + 2k\pi}{n}\right)\]\For the fourth roots, \(n = 4\): \[16^{1/4} = \sqrt[4]{16} = 2\]\Now calculate each root for \(k = 0, 1, 2, 3\): 1. \(k=0\): \(2 \text{{cis}} \left(\frac{5\pi}{24}\right)\)\2. \(k=1\): \(2 \text{{cis}} \left(\frac{11\pi}{24}\right)\)\3. \(k=2\): \(2 \text{{cis}} \left(\frac{17\pi}{24}\right)\)\4. \(k=3\): \(2 \text{{cis}} \left(\frac{23\pi}{24}\right)\)

Step 3: Convert Back to Rectangular Form

Translate the polar forms back into rectangular form: \1. \(z_1 = 2(\cos(\frac{5\pi}{24}) + i \sin(\frac{5\pi}{24}))\)\2. \(z_2 = 2(\cos(\frac{11\pi}{24}) + i \sin(\frac{11\pi}{24}))\)\3. \(z_3 = 2(\cos(\frac{17\pi}{24}) + i \sin(\frac{17\pi}{24}))\)\4. \(z_4 = 2(\cos(\frac{23\pi}{24}) + i \sin(\frac{23\pi}{24}))\)\These can be calculated with a calculator to find the approximate values of each root.

Key Concepts

Polar Form

The polar form of a complex number offers a convenient way to represent it using magnitude and direction. Instead of using the standard format with a real and an imaginary part, the polar form describes a complex number in terms of:

• its modulus \(r\), which is the distance from the origin to the point in the complex plane, and
• an angle \(\theta\), which indicates the direction from the positive real axis.

To find the modulus, we use the formula \( r = \sqrt{x^2 + y^2} \), where \(x\) and \(y\) represent the real and imaginary parts, respectively.
For example, given the complex number \(-8 + 8\sqrt{3} i\), the modulus is 16. The angle \(\theta\) is calculated using the tangent function, where \(\tan(\theta) = \frac{y}{x}\). Finding the correct quadrant for \(\theta\) is vital. Here, the negative real part places our number in the second quadrant, so \(\theta = \frac{5\pi}{6}\). Hence, the polar form is represented concisely as \(16 \text{cis} \frac{5\pi}{6}\), where \(\text{cis}\) is shorthand for cosine and sine combined as \(\cos(\theta) + i\sin(\theta)\).

De Moivre's Theorem

De Moivre's Theorem provides a powerful method for calculating powers and roots of complex numbers. For a complex number in polar form \(r \text{cis} \theta\), the theorem states that its \(n\)-th power or root can be computed through the formula:

• \((r^{1/n}) \text{cis} \left( \frac{\theta + 2k\pi}{n} \right)\)

Here, \(k\) is an integer that helps identify the various roots, particularly when \(n\) is greater than one.In our example, to calculate the fourth roots of the original number, we first determine \(r^{1/4} = 2\). Using De Moivre’s Theorem, we then compute four distinct roots by substituting \(k\) values from 0 to 3 into our formula, producing the roots:

• \(2 \text{cis} \left( \frac{5\pi}{24} \right)\)
• \(2 \text{cis} \left( \frac{11\pi}{24} \right)\)
• \(2 \text{cis} \left( \frac{17\pi}{24} \right)\)
• \(2 \text{cis} \left( \frac{23\pi}{24} \right)\)

These representations are geometrically spaced evenly around the circle in the complex plane, showing the symmetry of root distribution.

Rectangular Form

The rectangular form of a complex number expresses it using real and imaginary parts. While polar form is great for multiplying and finding roots, rectangular form is practical for addition, subtraction, and visualization. The conversion between the two forms can adapt flexibly depending on what we need to do with the number.To transform from polar back to rectangular form, use trigonometric expressions:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

This allows us to write a complex number as \(x + yi\). For example, converting the four roots we found into rectangular form involves calculating the cosine and sine for each angle \(\theta\), and then performing multiplication with \(r = 2\).After doing the math, it turns out that each polar expression \(2 \text{cis} \theta\) has a corresponding rectangular form such as:

• \(z_1 = 2(\cos(\frac{5\pi}{24}) + i \sin(\frac{5\pi}{24}))\)
• \(z_2 = 2(\cos(\frac{11\pi}{24}) + i \sin(\frac{11\pi}{24}))\)
• \(z_3 = 2(\cos(\frac{17\pi}{24}) + i \sin(\frac{17\pi}{24}))\)
• \(z_4 = 2(\cos(\frac{23\pi}{24}) + i \sin(\frac{23\pi}{24}))\)

Using a calculator for trigonometric evaluations will yield the numerical approximations for these forms, showing their unique real and imaginary components.