Problem 21 - Contemporary Precalculus

Question

Find the nth roots in polar form. $$81\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right) ; \quad n=4$$

Step-by-Step Solution

Answer

Answer: The four fourth roots of the given complex number in polar form are: 1. \(z_0 = 3\left(\cos \frac{\pi}{12} + i \sin \frac{\pi}{12}\right)\) 2. \(z_1 = 3\left(\cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12}\right)\) 3. \(z_2 = 3\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\) 4. \(z_3 = 3\left(\cos \frac{13\pi}{12} + i \sin \frac{13\pi}{12}\right)\)

Step 1: Express the given complex number in polar form

The complex number is already given in its polar form: $$z = 81\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$$ where the modulus \(r = 81\) and the argument \(\theta = \frac{\pi}{12}\).

Step 2: Apply De Moivre's theorem

To find the nth roots of a complex number in polar form, we apply De Moivre's theorem: $$z_k = r^{\frac{1}{n}}\left(\cos{\frac{\theta + 2k\pi}{n}} + i\sin{\frac{\theta + 2k\pi}{n}}\right)$$ for \(k = 0, 1, ..., n-1\). In this case, \(n = 4\), \(r = 81\), and \(\theta = \frac{\pi}{12}\).

Step 3: Express the fourth roots as complex numbers in polar form

Now, let's compute the fourth roots by substituting \(k = 0, 1, 2, 3\) into the formula above: $$z_0 = 3\left(\cos \frac{\pi}{12} + i \sin \frac{\pi}{12}\right)$$ $$z_1 = 3\left(\cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12}\right)$$ $$z_2 = 3\left(\cos \frac{9\pi}{12} + i \sin \frac{9\pi}{12}\right) = 3\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)$$ $$z_3 = 3\left(\cos \frac{13\pi}{12} + i \sin \frac{13\pi}{12}\right)$$ These are the four fourth roots of the given complex number in polar form: 1. \(z_0 = 3\left(\cos \frac{\pi}{12} + i \sin \frac{\pi}{12}\right)\) 2. \(z_1 = 3\left(\cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12}\right)\) 3. \(z_2 = 3\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\)  4. \(z_3 = 3\left(\cos \frac{13\pi}{12} + i \sin \frac{13\pi}{12}\right)\)

Key Concepts

De Moivre's Theorem

De Moivre's theorem is a powerful tool in complex number analysis, particularly for raising complex numbers to integer powers and extracting roots. For any complex number expressed in polar form as \( z = r(\text{cos}θ + i\text{sin}θ) \), where \( r \) is the modulus (distance from the origin in the complex plane) and \( θ \) is the argument (the angle with the positive real axis), De Moivre's theorem states that:

\[ z^n = r^n(\text{cos}nθ + i\text{sin}nθ) \]
This formula simplifies the computation of powers of complex numbers. When extracting nth roots, you would take the nth root of the modulus and divide the argument by \( n \). Each root is then calculated by adding \( \frac{2kπ}{n} \) to the argument, where \( k \) is an integer from 0 to \( n-1 \), representing the different roots. Thus, the formula for the nth roots of \( z \) is:

\[ z_k = r^{\frac{1}{n}}\left(\text{cos}\frac{θ + 2kπ}{n} + i\text{sin}\frac{θ + 2kπ}{n}\right) \]
De Moivre's theorem is not only simple and elegant but also a key concept in the field of complex analysis, as it connects trigonometry and complex numbers in an intuitive way.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to a two-dimensional complex plane by including the square root of negative one, known as \( i \), the imaginary unit. A complex number has a 'real' part and an 'imaginary' part and is written as \( a + bi \), where \( a \) and \( b \) are real numbers. Complex numbers can be added, subtracted, multiplied, and divided, obeying specific algebraic rules.

The beauty of complex numbers lies in their ability to provide a coherent framework for solving equations that have no solution in the real number system, such as \( x^2 + 1 = 0 \). Their properties are essential in various fields, including engineering, physics, and applied mathematics, by allowing for a comprehensive description of oscillations, waves, and other phenomena.

Polar Form of Complex Numbers

The polar form is another way to represent complex numbers, focusing on their magnitude and direction rather than their position along the standard, real, and imaginary axes. In the polar form, a complex number \( z = a + bi \) is written as:

\[ z = r(\text{cos}θ + i\text{sin}θ) \]
Here, \( r \) is the modulus of \( z \), calculated as \( \sqrt{a^2 + b^2} \), and \( θ \) is the argument, the angle formed with the positive real axis, which can be determined using the arctangent of \( b/a \). The polar form is particularly helpful when multiplying or dividing complex numbers as these operations become simplified to multiplying their moduli and adding or subtracting their arguments.

When we put complex numbers into polar form, we gain a new perspective that is particularly useful in many areas of mathematics and physics. It makes visualizing complex multiplication and division as simple geometric transformations, such as rotations and dilations, within the complex plane.