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Chapter 11: Problem 38

Find all indicated roots and express them in rectangular form. Check your results with a calculator. The fourth roots of \(-1\).

Short Answer

Expert verified
The fourth roots of -1 are \(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\), \(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\), \(-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\), and \(\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).

Step by step solution

01

Express in Polar Form

To find the fourth roots of -1, you'll first want to express -1 in polar form. The complex number -1 can be expressed as 1 (magnitude) angle\(\pi\). So in polar form, it's 1 angle\(\pi\): \(re^{i\theta} = e^{i\pi} \).
02

Use De Moivre's Theorem

Use De Moivre鈥檚 Theorem, which says that the n-th roots of a complex number are given by:\(z^{1/n} = r^{1/n} (\cos(\frac{\theta + 2k\pi}{n}) + i \sin(\frac{\theta + 2k\pi}{n}))\), where k = 0, 1, ..., n-1.
03

Calculate Magnitude

For the fourth root problem, n=4, \(r = 1\), thus the magnitude \(r^{1/4} = 1^{1/4} = 1\), because any non-zero number raised to any power remains 1.
04

Calculate Angles

We need to find all four roots, so compute angles for k = 0, 1, 2, 3 using:\(\theta_k = \frac{\pi + 2k\pi}{4}\).
05

Calculate for k = 0

For k = 0: \(\theta_0 = \frac{\pi}{4}\),so the first root is \(e^{i\frac{\pi}{4}} = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\).
06

Calculate for k = 1

For k = 1: \(\theta_1 = \frac{3\pi}{4}\), thus the second root is \(e^{i\frac{3\pi}{4}} = \cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\).
07

Calculate for k = 2

For k = 2: \(\theta_2 = \frac{5\pi}{4}\), so the third root is \(e^{i\frac{5\pi}{4}} = \cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).
08

Calculate for k = 3

For k = 3: \(\theta_3 = \frac{7\pi}{4}\), thus the fourth root is \(e^{i\frac{7\pi}{4}} = \cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4} = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Form
Polar form is a way to represent complex numbers using a magnitude and an angle. Instead of using the standard "a + bi" format known as rectangular form, polar form expresses a complex number as a modulus (r) and an angle (胃).

A complex number like \(-1\) can be rewritten in polar form as \(re^{i\theta}\). In polar form, \(-1\) has a magnitude of 1 (distance from origin), and its angle with respect to the positive x-axis is \(\pi\) radians or 180 degrees.
  • The magnitude \(r\) shows the distance from the origin to the point in the complex plane.
  • The angle \(\theta\) indicates the counterclockwise rotation from the positive real axis.
So, \(-1\) in polar form is expressed as \(e^{i\pi}\). This conversion is crucial for problems like finding roots, as it simplifies expressions and calculations.
De Moivre's Theorem
De Moivre's Theorem provides a straightforward method for finding roots and powers of complex numbers. It states that if a complex number is expressed in polar form as \(re^{i\theta}\), then the nth roots are given by:

\[ z^{1/n} = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \]
  • "n" is the number of roots you are seeking, so for fourth roots, \(n=4\).
  • "k" is an integer that runs from 0 to \(n-1\), which allows calculation of all n roots.
To find the fourth roots of \(-1\), De Moivre鈥檚 theorem dictates computing angles for each \(k = 0, 1, 2, \text{and}\ 3\). Hence, the angles are calculated using: \(\theta_k = \frac{\pi + 2k\pi}{4}\). The method lets us convert angles conveniently back to trigonometric terms, aiding in finding each root.
Rectangular Form
Rectangular form, also known as Cartesian form, is the standard format to express complex numbers. It is written as \(a + bi\), where:
  • "a" denotes the real part
  • "b" is the imaginary part
In the rectangular form, each complex number corresponds to a point on the complex plane, using the x-y coordinate system where "a" is along the x-axis (real) and "b" is along the y-axis (imaginary).

Once you've used De Moivre's Theorem to find the roots in polar form, it's often necessary to convert them back to rectangular form for a more familiar interpretation. For each root:
  • Convert the polar angle to its corresponding trigonometric values, using cosine for the real part and sine for the imaginary part.
  • For example, the root \(e^{i\frac{\pi}{4}}\) converts to \(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\).
This form is especially useful when verifying your results with software or calculators, which often use rectangular form as a standard output.

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Most popular questions from this chapter

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=2 \sin \theta$$

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). (GRAPH CANNOT COPY) $$x=2 \sin 2 t, y=3 \cos 3 t$$

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Radio stations sometimes do not broadcast in all directions with the same intensity. To avoid interference with an existing station to the north, a new station may be licensed to broadcast only east and west. To create an east- west signal, two radio towers are used, as illustrated in the figure. (IMAGE CAN'T COPY). Locations where the radio signal is received correspond to the interior of the lemniscate..$$r^{2}=40,000 \cos 2 \theta$$.Where the polar axis (or positive \(x\) -axis) points east. (A) Graph $$r^{2}=40.000 \cos 2 \theta$$. For \(0^{\circ} \leq \theta \leq 180^{\circ},\) with units in miles. Assuming that the radio towers are located near the pole, use the graph to describe the regions where the signal can be received and where the signal cannot be received. (B) Suppose a radio signal pattern is given by $$r^{2}=22,500 \sin 2 \theta$$ for \(0^{\circ} \leq \theta \leq 180^{\circ} .\) Graph this pattem and interpret the results.
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