Problem 137 - 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. Square roots of \(5\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\)

Step-by-Step Solution

Answer

The square roots of \(5\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\) are given by the points on the graph computed in Step 2 and in standard form as calculated in Step 3.

Step 1: Applying De Moivre's Theorem

We start by applying De Moivre's Theorem, which states that if \(z=r\left(\cos \theta+i \sin \theta\right)\), then the \(n\)th roots of \(z\) are given by: \(r^{1 / n}\left[\cos \left(\frac{\theta+2 k \pi}{n}\right)+\sin \left(\frac{\theta+2 k \pi}{n}\right) i\right]; k=0,1,2, \ldots, n-1.\) For this problem, we have \(n=2\), \(r=5\) and \(\theta = 120^\circ\). Substitute these values into the formula and calculate the roots.

Step 2: Visualization of Roots on Graph

Take the roots from the previous step and plot them on the Argand plane, or complex plane. Each root will be represented as a point in the system, with \(x\) coordinate being the real part and \(y\) coordinate being the imaginary part of each root.

Step 3: Converting Roots to Standard Form

Take the square roots obtained from Step 1, which are in Polar form, and convert to Standard form (a + bi). Here, \(a\) is the real part and \(b\) is the imaginary part. They can be calculated by using the expressions: \(a = r \cos \theta\) and \(b = r \sin \theta\). Apply them to each root.

Key Concepts

Complex Numbers

Complex numbers are indispensable in mathematics, acting as extensions of real numbers. They come in the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). This structure allows them to represent two-dimensional numbers in a way that captures both magnitude and direction.
Complex numbers can be visualized on the complex plane, much like how real numbers are visualized on a number line. This plane includes:

• The real axis, where the real part \( a \) is placed.
• The imaginary axis, where the imaginary part \( b \) is plotted.

Understanding complex numbers is key, as they are the foundation for operations in both polar and rectangular forms, as demonstrated by their diverse applications in solving polynomial equations and representing periodic systems.

Polar Form

Polar form represents complex numbers in terms of their magnitude and angle. This representation is particularly useful when dealing with multiplication, division, and finding powers and roots of complex numbers, as it simplifies calculations significantly. A complex number \( z = a + bi \) can be transformed into polar form via\[ z = r (\cos \theta + i \sin \theta) \]where:

• \( r \) is the magnitude or modulus, calculated as \( r = \sqrt{a^2 + b^2} \)
• \( \theta \) is the argument or angle, determined by \( \theta = \tan^{-1}(\frac{b}{a}) \)

In polar form, complex numbers are often expressed as \( r \text{cis} \theta \), where \( \text{cis} \theta \) combines the cosine and sine terms. De Moivre's Theorem, crucially important for calculating powers and roots, becomes remarkably straightforward when using this representation.

Standard Form

Standard form, or rectangular form, of a complex number is the basic \( a + bi \) expression. Converting from polar to standard form requires using the relationships:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

For instance, after using De Moivre’s Theorem to find roots, results initially in polar coordinates must be converted into this form for ease of interpretation and calculation in many contexts. This transition enables visualization on the complex plane or application in functions needing rectangular inputs. Understanding both the polar and standard forms is essential to fully harness the versatility of complex numbers.

Roots of Complex Numbers

Finding the roots of complex numbers, especially non-real ones, can appear daunting, but using De Moivre's Theorem simplifies the process. The theorem states that to find the \( n \)th roots of a complex number \( z = r \text{cis} \theta \), you apply:\[ r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right) \]where \( k = 0, 1, 2, \ldots, n-1 \) and provides \( n \) distinct roots evenly distributed around the origin in the complex plane. This method balances the foundational concept of symmetry in complex roots, offering equal angular spacing, which is often graphically represented to provide insight into their geometric relationship.
For example, roots extracted from \( 5(\cos 120^{\circ} + i \sin 120^{\circ}) \) using De Moivre’s involve setting \( n = 2 \), yielding roots visually identifiable on the Argand diagram. This approach not only reveals the precise location of roots but also unlocks solutions to equations beyond the reach of simple algebraic methods.