Problem 139 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use the theorem on page 590 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Fourth roots of \(81\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\)

Step-by-Step Solution

Answer

The fourth roots of \(81(\cos \frac{2\pi}{3} + i\sin \frac{2\pi}{3})\) are \(3(\cos (\frac{2\pi}{3} + i\sin \frac{2\pi}{3}))\), \(3(\cos (\frac{5\pi}{3} + i\sin \frac{5\pi}{3}))\), \(3(\cos (\frac{8\pi}{3} + i\sin \frac{8\pi}{3}))\), and \(3(\cos (\frac{11\pi}{3} + i\sin \frac{11\pi}{3}))\). These can be represented in standard form as \(3e^{i2\pi/3}\), \(3e^{i5\pi/3}\), \(3e^{i8\pi/3}\), and \(3e^{i11\pi/3}\).

Step 1: Application of De Moivre's Theorem

Using De Moivre's theorem, \( (r(\cos \theta + i\sin \theta))^{n} = r^{n}(\cos n\theta + i\sin n\theta) \), we can find the fourth roots of the complex number \(81(\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3})\). The fourth roots of the complex number will be \(81^{1/4}(\cos (\frac{2\pi}{3} + 2k\pi) + i\sin (\frac{2\pi}{3}+ 2k\pi))\), where \(k = 0, 1, 2, 3 \).

Step 2: Calculation of Fourth Roots

Substitute \(k = 0, 1, 2, 3\) into the equation to get four fourth roots: For \(k = 0\), \(3(\cos (\frac{2\pi}{3}) + i\sin (\frac{2\pi}{3}))\), For \(k = 1\), \(3(\cos (\frac{5\pi}{3}) + i\sin (\frac{5\pi}{3}))\), For \(k = 2\), \(3(\cos (\frac{8\pi}{3}) + i\sin (\frac{8\pi}{3}))\), For \(k = 3\), \(3(\cos (\frac{11\pi}{3}) + i\sin (\frac{11\pi}{3}))\).

Step 3: Graphical Representation

The roots should be represented on a 2D coordinate plane, where the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part of the complex roots. The roots will be evenly spaced around the circle of radius 3 in the complex plane.

Step 4: Conversion to Standard Form

The roots can then be converted to standard form using the definitions of cosine and sine in terms of the exponential function. The roots are: \(3e^{i2\pi/3}\), \(3e^{i5\pi/3}\), \(3e^{i8\pi/3}\), \(3e^{i11\pi/3}\).

Key Concepts

complex number roots

When working with complex numbers, the concept of roots is crucial. Complex number roots refer to solutions of an equation involving complex numbers. Using De Moivre's Theorem is an effective method to find these roots. For a complex number in polar form, such as \( z = r (\cos \theta + i \sin \theta) \), calculating the \( n \)-th roots involves using the formula:

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

Here, \( r \) refers to the modulus and \( \theta \) is the argument of the complex number. The integer \( k \) takes values from 0 up to \( n-1 \) to generate all distinct \( n \)-th roots. For example, to find the fourth roots of a complex number, you would substitute \( k = 0, 1, 2, 3 \). This allows for the calculation of each root positioned at equal intervals on the unit circle. Understanding these roots involves both numerical calculations and a deep appreciation of geometric patterns in the complex plane.

graphical representation of complex numbers

Visualizing complex numbers is an integral part of grasping their behavior. This is often done on the complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. When representing the roots of a complex number, each root can be plotted as a point in this plane.For instance, when calculating the fourth roots of a given complex number, those roots will be positioned equidistantly on a circle with a radius equal to the modulus of the roots. In our specific example of fourth roots, each will lie on a circle of radius 3, due to the value of \( 81^{1/4} = 3 \). These points form vertices of a regular polygon, like a square for fourth roots, since the angle differences between consecutive roots are \( \frac{2\pi}{4} = \frac{\pi}{2} \) radians.Graphical representation aids in appreciating the symmetry and rotational properties inherent in complex roots. It also offers insights into the nature of the polynomial equations solved by such roots.

conversion to standard form

Converting complex numbers from polar to standard form involves expressing them as \( a + bi \), where \( a \) and \( b \) are real numbers signifying the real and imaginary parts, respectively. This conversion uses the identities of sine and cosine, which relate to the exponential function:

• \( \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \)
• \( \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \)

After calculating the roots in exponential form, using De Moivre's theorem, convert

• \( 3(\cos \theta + i\sin \theta) \text{ into } 3(\frac{e^{i\theta} + e^{-i\theta}}{2} + i \frac{e^{i\theta} - e^{-i\theta}}{2i}) \)

Expanding and simplifying the expression yields the numbers in the familiar \( a + bi \) form.For each root calculated, repeat this process to depict them all in standard form. This traditional representation is crucial for use in further calculations and understanding their roles within polynomial equations.