Problem 83 - Precalculus

Question

Refer to the following: According to the \(n\) th root theorem, the first of the \(n\) th roots of the complex number \(z=r(\cos \theta+i \sin \theta)\) is given by \(w_{1}=r^{1 / n}\left[\cos \left(\frac{\theta}{n}+\frac{2 \pi}{n}\right)+i \sin \left(\frac{\theta}{n}+\frac{2 \pi}{n}\right)\right],\) with \(\theta\) in radians or \(w_{1}=r^{1 / n}\left[\cos \left(\frac{\theta}{n}+\frac{360^{\circ}}{n}\right)+i \sin \left(\frac{\theta}{n}+\frac{360^{\circ}}{n}\right)\right]\) with \(\theta\) in degrees. Using the graphing calculator to plot the \(n\) roots of a complex number \(z,\) enter \(r_{1}=r, \theta \min =\frac{\theta}{n}, \theta \max =2 \pi+\frac{\theta}{n}\) or \(360^{\circ}+\frac{\theta}{n}, \theta\) step \(=\frac{2 \pi}{n}\) or \(\frac{360^{\circ}}{n}, \mathrm{xmin}=-r, \mathrm{xmax}=r\) ymin \(=-r,\) ymax \(=r,\) and \([\mathrm{MODE}]\) in radians or degrees. Find the fifth roots of \(\frac{\sqrt{3}}{2}-\frac{1}{2} i,\) and plot the roots with a calculator.

Step-by-Step Solution

Answer

The fifth roots of \( \frac{\sqrt{3}}{2} - \frac{1}{2} i \) are \( w_0, w_1, w_2, w_3, \) and \( w_4 \). Use a graphing calculator to plot them as described.

Step 1: Convert Complex Number to Trigonometric Form

Given complex number is \( \frac{\sqrt{3}}{2} - \frac{1}{2}i \). To convert it to the form \( r(\cos \theta + i \sin \theta) \), find \( r = |z| \) and \( \theta \).Calculate \( r \):\[ r = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]Calculate \( \theta \):Using \( \cos \theta = \frac{\sqrt{3}}{2} \) and \( \sin \theta = -\frac{1}{2} \), we find \( \theta = -\frac{\pi}{6} \) in radians (or \( -30^\circ \) in degrees) since it is in the 4th quadrant.

Step 2: Apply the Fifth Root Formula

Use the formula for the fifth roots: \[ w_k = r^{1/5} \left[\cos\left(\frac{\theta + 2k\pi}{5}\right) + i \sin\left(\frac{\theta + 2k\pi}{5}\right)\right] \]where \( k = 0, 1, 2, 3, 4 \), \( r = 1 \), and \( \theta = -\frac{\pi}{6} \).

Step 3: Calculate the Fifth Roots

For each value of \( k \), calculate \( w_k \):1. \( w_0 = 1^{1/5} \left[\cos\left(\frac{-\pi/6 + 0 \cdot 2\pi}{5}\right) + i\sin\left(\frac{-\pi/6 + 0 \cdot 2\pi}{5}\right)\right] = \cos\left(-\frac{\pi}{30}\right) + i \sin\left(-\frac{\pi}{30}\right) \)2. \( w_1 = \cos\left(\frac{-\pi/6 + 2\pi}{5}\right) + i \sin\left(\frac{-\pi/6 + 2\pi}{5}\right) \)3. \( w_2 = \cos\left(\frac{-\pi/6 + 4\pi}{5}\right) + i \sin\left(\frac{-\pi/6 + 4\pi}{5}\right) \)4. \( w_3 = \cos\left(\frac{-\pi/6 + 6\pi}{5}\right) + i \sin\left(\frac{-\pi/6 + 6\pi}{5}\right) \)5. \( w_4 = \cos\left(\frac{-\pi/6 + 8\pi}{5}\right) + i \sin\left(\frac{-\pi/6 + 8\pi}{5}\right) \)

Step 4: Plot the Roots Using a Calculator

Set the graphing calculator to mode in radians and input the following settings based on \( r = 1 \):- \( r_1 = 1 \)- \( \theta_{\min} = \frac{-\pi}{30} \)- \( \theta_{\max} = \frac{-\pi}{30} + 2\pi \)- \( \theta \text{ step} = \frac{2\pi}{5} \)- \( \mathrm{xmin} = -1, \mathrm{xmax} = 1 \)- \( \mathrm{ymin} = -1, \mathrm{ymax} = 1 \)Plot points for each calculated fifth root.

Key Concepts

Understanding Complex Numbers

Complex numbers are essential in mathematics and engineering. They consist of a real part and an imaginary part, represented as:

• Real part, often denoted by "a"
• Imaginary part, denoted by "bi" where "i" is the square root of -1.

For example, in the complex number \( \frac{\sqrt{3}}{2} - \frac{1}{2}i \), \

1. The real part is \( \frac{\sqrt{3}}{2} \)
2. The imaginary part is \( -\frac{1}{2} \)}

It's useful to visualize them as points or vectors in a 2D coordinate system: the x-axis is the real number line, and the y-axis represents the imaginary part. Each complex number corresponds to a unique point in this plane.

The Trigonometric Form of Complex Numbers

Converting complex numbers to trigonometric form simplifies operations like multiplication and finding roots. This form is expressed as \( r(\cos \theta + i \sin \theta) \), where:

• \( r \) is the modulus, representing the distance from the origin to the point \((a, b)\).
• \( \theta \) is the argument, which is the angle formed with the positive x-axis.

To convert the complex number \( \frac{\sqrt{3}}{2} - \frac{1}{2}i \):

1. Calculate \( r = 1 \), as shown in the original solution.
2. Determine \( \theta = -\frac{\pi}{6} \), derived from the known values of sine and cosine ratios in the 4th quadrant.

This trigonometric form helps in understanding the geometry of complex numbers, making further calculations more straightforward.

Plotting with a Calculator

Graphing calculators are incredibly useful for visualizing complex number operations, especially when finding roots. Here's a basic setup guide:

• Set your calculator to radian mode for trigonometric calculations.
• Input parameters: \( r = 1 \), \( \theta_{\min} = \frac{-\pi}{30} \), \( \theta_{\max} = 2\pi + \frac{-\pi}{30} \).
• Define the step for theta: \( \theta \text{ step} = \frac{2\pi}{5} \).
• Set axes: \( \mathrm{xmin} = -1, \mathrm{xmax} = 1, \mathrm{ymin} = -1, \mathrm{ymax} = 1 \).

Once these settings are input, your calculator will plot the complex roots as points on the complex plane. Observing these plots can help better understand the distribution and symmetry of complex number roots.

Diving Deeper into Fifth Roots

The fifth roots of a complex number can be thought of as evenly spaced points on the circle defined by the modulus in the complex plane. For the trigonometric form, we apply the formula: \[ w_k = r^{1/5} \left[ \cos\left(\frac{\theta + 2k\pi}{5}\right) + i \sin\left(\frac{\theta + 2k\pi}{5}\right) \right] \] where \( k = 0, 1, 2, 3, 4 \). This might seem daunting, but breaking it down:

• For each \( k \), the angle changes by \( \frac{2\pi}{5} \), spacing them equally on the circle.
• The modulus after taking the fifth root remains 1, simplifying the calculations.

These points represent all possible fifth roots and form a regular pentagon when plotted, thanks to their uniform angular separation. Recognizing these geometric properties highlights the elegance and symmetry inherent in complex numbers.