Question

Follow the procedure illustrated in Example 4 to determine the indicated roots of the given complex number. $$ [2(\cos \pi / 3+i \sin \pi / 3)]^{1 / 2} $$

Short answer

Question: Calculate the square root of the complex number \(2(\cos \pi / 3+i \sin \pi / 3)\). Answer: The square roots of the given complex number are \(\sqrt[\frac{1}{2}]{2} (\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})\) and \(\sqrt[\frac{1}{2}]{2} (\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3})\).

Step-by-step solution

Identify r, θ, and n

First, let's identify the values for \(r\), \(\theta\), and \(n\) in our complex number. From the given complex number, we have: \(r = 2\) \(\theta = \pi/3\) \(n = 1/2\)

Use De Moivre's Theorem formula

We will now apply De Moivre's theorem with the established values. For \(k = 0\): $$ \sqrt[\frac{1}{2}]{2} (\cos \frac{\pi / 3 + 2(0)\pi}{1 / 2} + i \sin \frac{\pi / 3 + 2(0)\pi}{1 / 2}) $$ For \(k = 1\): $$ \sqrt[\frac{1}{2}]{2} (\cos \frac{\pi / 3 + 2(1)\pi}{1 / 2} + i \sin \frac{\pi / 3 + 2(1)\pi}{1 / 2}) $$

Calculate the roots

Now we will find the two roots. Root 1 (k=0): $$ \sqrt[\frac{1}{2}]{2} (\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) $$ Root 2 (k=1): $$ \sqrt[\frac{1}{2}]{2} (\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}) $$

Write the final answer

The two roots of the given complex number are: Root 1: $$ \sqrt[\frac{1}{2}]{2} (\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) $$ Root 2: $$ \sqrt[\frac{1}{2}]{2} (\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}) $$

Key concepts

De Moivre's Theorem

De Moivre's Theorem elegantly connects complex numbers and trigonometry by providing a formula for raising complex numbers to integer and fractional powers. The theorem is expressed as:
\[ (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \]
where:

• r is the modulus of the complex number
• θ is the argument, or angle
• n is the exponent

By using De Moivre's theorem, one can easily compute powers and roots of complex numbers when they are expressed in polar form. In the case of fractional powers, such as determining roots, the theorem requires calculating each possible value of k, where k represents each distinct root. It simplifies the otherwise complex operations through trigonometric functions.

Complex Roots

Finding complex roots refers to the process of discovering multiple values that satisfy an equation involving a complex number, usually when dealing with powers and roots. For example, finding the square roots of a complex number involves finding two different solutions, which may not only have real components but also imaginary ones.

When applying De Moivre's Theorem to find complex roots, you'll deal with the parameter k in a cyclic fashion. For a root of order \( n \), you'll explore solutions for \( k = 0, 1, 2, ..., n-1 \). Each results yields a different angle and thus a distinct root. This cyclical nature reflects the periodicity of trigonometric functions in the complex plane, offering a complete set of roots, which in the context of square roots, typically results in two solutions, as shown in the exercise.

Polar Form

Polar form is an alternative way to express complex numbers. Instead of using the typical Cartesian coordinates \( x + yi \), we use a magnitude and an angle, represented as \( r(\cos \theta + i \sin \theta) \), which can also be simplified as \( r \text{cis} \theta \).

In this context:

• r is known as the modulus or absolute value of the complex number, denoting its distance from the origin in the complex plane.
• \(\theta\) is the argument, representing the direction measured as the angle from the positive real axis.

Using polar form allows for more intuitive operations on complex numbers, like multiplication and finding roots. This is particularly useful in trigonometric functions due to their cyclical nature.

Trigonometric Form

The trigonometric form of a complex number is a direct expression involving sines and cosines to reflect its position in the complex plane. It's closely linked to the polar form and is given by \( r(\cos \theta + i \sin \theta) \).

This form highlights:

• Cosine \(\cos \theta\) denotes the horizontal component, or the real part, of the complex number.
• Sine \(\sin \theta\) expresses the vertical component, or the imaginary part.

The trigonometric form provides a natural connection to various applications in fields such as physics and engineering, where oscillations and rotations are frequent. It simplifies computation with complex numbers, making it especially beneficial for applying identities and theorems like De Moivre's.