Problem 143 - 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. Cube roots of \(-\frac{125}{2}(1+\sqrt{3} i)\)

Step-by-Step Solution

Answer

The cube roots of the given complex number are \(\frac{5}{2} + i \frac{5\sqrt{3}}{2}\), -5, and \(\frac{5}{2} - i \frac{5\sqrt{3}}{2}\).

Step 1: Convert to Polar Form

Given complex number is \(-\frac{125}{2}(1+\sqrt{3} i)\). The modulus of the complex number \(r = \sqrt{(\frac{-125}{2})^2 + (\frac{125\sqrt{3}}{2})^2} = 125\). The argument (in radians) \(\theta = \tan^{-1}(\frac{-\frac{125\sqrt{3}}{2}}{-\frac{125}{2}}) = \frac{1\pi}{3}\). Therefore, the polar form of the given complex number is \(125 (\cos \frac{1\pi}{3} + i \sin \frac{1\pi}{3} )\).

Step 2: Apply De Moivre's Theorem

According to the De Moivre's theorem, the cube roots of the complex number will be \(r^{1/3} (\cos\frac{1\pi +2k\pi}{3}+i \sin\frac{1\pi+2k\pi}{3})\), where \(k = 0, 1, 2\). Substituting \(r^{1/3} = \sqrt[3]{125} = 5\), our roots are for \(k = 0\) - \(A = 5 (\cos\frac{\pi }{3}+i \sin\frac{\pi }{3})\), for \(k = 1\) - \(B = 5 (\cos\pi +i \sin\pi)\) and for \(k = 2\) - \(C = 5 (\cos\frac{5\pi }{3}+i \sin\frac{5\pi }{3})\).

Step 3: Convert to Rectangular Form

Finally, convert the obtained cube roots into rectangular form. \(A = 5 (cos\frac{\pi}{3} + i sin\frac{\pi}{3}) = 5(\frac{1}{2} + i \frac{\sqrt{3}}{2}) = \frac{5}{2} + i \frac{5\sqrt{3}}{2}\), \(B = 5 (cos\pi + i sin\pi ) = -5\), and \(C = 5 (cos\frac{5\pi}{3} + i sin\frac{5\pi}{3}) = 5(\frac{1}{2} - i \frac{\sqrt{3}}{2}) = \frac{5}{2} - i \frac{5\sqrt{3}}{2}\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a formula that connects complex numbers with trigonometry. It enables mathematicians and students to raise complex numbers to any power using a simple expression involving trigonometric functions.

Specifically, if you have a complex number in polar form, which is expressed as \( r(\text{cos} \theta + i\text{sin} \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument), De Moivre's Theorem says that this complex number raised to an integer power \( n \) is given by:

\[ r^n (\text{cos}(n\theta) + i\text{sin}(n\theta)) \]

When finding the roots of a complex number, as seen in the exercise, De Moivre's Theorem provides a systematic way to find all solutions. For cube roots (\( n = 3 \)), you would divide the argument by 3 and multiply by \( k\cdot2\pi \) for \( k \) ranging from 0 to 2 to get each distinct root.

This theorem simplifies what could be a daunting exponential problem into manageable trigonometric expressions.

Polar Form of Complex Numbers

The polar form of complex numbers is an alternative representation that uses a combination of an angle and a distance to describe a number in the complex plane, rather than the traditional 'x' and 'y' coordinates.

Any complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers, can be represented in polar form as:

\[ r(\text{cos} \theta + i\text{sin} \theta) \]
where

• \( r \) is the modulus of \( z \), calculated as \( r = \sqrt{a^2 + b^2} \)
• \( \theta \) is the argument of \( z \), which is the angle formed with the positive real axis, calculated as \( \theta = \text{arg}(z) = \text{tan}^{-1}(\frac{b}{a}) \)

The conversion to polar form is crucial when applying De Moivre's Theorem since it sets the stage for transforming the complex number into a more workable formula for exponentiation and root extraction, as it directly shows the complex number's magnitude and direction.

Rectangular Form of Complex Numbers

The rectangular form of complex numbers is the standard and most commonly used way to represent complex numbers. In this form, a complex number is expressed as the sum of real and imaginary parts, which can be plotted on a traditional two-dimensional graph.

The general format for a complex number in rectangular form is:\[ z = a + bi \]

where

• \( a \) is the real part of the complex number, representing the horizontal axis coordinate,
• \( b \) is the imaginary part, representing the vertical axis coordinate, and \( i \) is the imaginary unit, satisfying the equation \( i^2 = -1 \).

The ease of visualization is one key advantage of the rectangular form. As seen in the original exercise's Step 3, once you have the polar form roots from De Moivre's Theorem, you can convert them back to the rectangular form using the corresponding sine and cosine values. This allows us to easily express complex numbers in a way that is familiar and convenient for algebraic operations such as addition and subtraction.