Problem 20 - Contemporary Precalculus

Question

Find the nth roots in polar form. $$8\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right) ; \quad n=3$$

Step-by-Step Solution

Answer

The 3rd roots of the complex number in polar form are: $$2\left(\cos \frac{\pi}{30} + i \sin \frac{\pi}{30}\right), \quad 2\left(\cos \frac{7\pi}{30} + i \sin \frac{7\pi}{30}\right), \quad 2\left(\cos \frac{13\pi}{30} + i \sin \frac{13\pi}{30}\right)$$

Step 1: Identify the given complex number in polar form

The given complex number is: $$8\left(\cos \frac{\pi}{10} + i \sin \frac{\pi}{10}\right)$$ Where the magnitude is 8 and the angle is \(\frac{\pi}{10}\).

Step 2: Apply De Moivre's theorem for nth roots

We need to find the 3rd roots, so n = 3. According to De Moivre's theorem, we will divide the angle by 3 and take the cube root of the magnitude. New magnitude: \(\sqrt[3]{8} = 2\) New angle: \(\frac{\pi}{10} \cdot \frac{1}{3} = \frac{\pi}{30}\) The first root has a magnitude of 2 and an angle of \(\frac{\pi}{30}\).

Step 3: Find the remaining roots

For the 3rd roots, there will be a total of 3 roots. We need to find the remaining 2 roots. To find the other roots, we will add multiples of \(\frac{2\pi}{3}\) to the angle of the first root. Second root angle: \(\frac{\pi}{30} + \frac{2\pi}{3} = \frac{7\pi}{30}\) Third root angle: \(\frac{\pi}{30} + 2\cdot\frac{2\pi}{3} = \frac{13\pi}{30}\)

Step 4: Write the 3rd roots in polar form

Now that we have the magnitudes and angles of each 3rd root, we can write them in polar form. First root: $$2\left(\cos \frac{\pi}{30} + i \sin \frac{\pi}{30}\right)$$ Second root: $$2\left(\cos \frac{7\pi}{30} + i \sin \frac{7\pi}{30}\right)$$ Third root: $$2\left(\cos \frac{13\pi}{30} + i \sin \frac{13\pi}{30}\right)$$ Therefore, the 3rd roots of the given complex number in polar form are: $$2\left(\cos \frac{\pi}{30} + i \sin \frac{\pi}{30}\right), \quad 2\left(\cos \frac{7\pi}{30} + i \sin \frac{7\pi}{30}\right), \quad 2\left(\cos \frac{13\pi}{30} + i \sin \frac{13\pi}{30}\right)$$

Key Concepts

Polar Form

Complex numbers can be represented in a form known as the polar form, which is particularly useful for multiplication and finding powers and roots.

The polar form of a complex number is expressed as:

• Magnitude (also known as modulus), which is the distance from the origin in the complex plane.
• Angle (also known as argument), which is the direction measured from the positive x-axis.

A complex number \( z \) can be expressed in polar form as:\[z = r \left( \cos \theta + i \sin \theta \right)\]Here, \( r \) is the magnitude, and \( \theta \) is the angle. This form is particularly efficient in problems involving powers of complex numbers.

When converting from the rectangular form (\( x + yi \)) to polar form, use:

• Magnitude: \( r = \sqrt{x^2 + y^2} \)
• Angle: \( \theta = \arctan \left( \frac{y}{x} \right) \)

De Moivre's Theorem

De Moivre's theorem is a powerful tool that connects complex numbers in polar form to trigonometry. It is especially useful for finding the powers and roots of complex numbers.

The theorem states:\[\left( r \left( \cos \theta + i \sin \theta \right) \right)^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right)\]This means that you raise the magnitude to the power of \( n \) and multiply the angle by \( n \). De Moivre's theorem simplifies the process of computing powers, and it also lays the groundwork for finding roots.

For example, in finding cube roots, you will:

• Divide the angle \( \theta \) by the root number.
• Take the root of the magnitude \( r \).

This method allows easy computation of roots and is foundational in many complex number problems.

Nth Roots of Complex Numbers

Finding the \( n \)th roots of a complex number is a systematic process requiring understanding of both polar form and De Moivre's theorem. If you have a complex number expressed in polar form \( r \left( \cos \theta + i \sin \theta \right) \), you can find the \( n \)th roots using the following steps:

1. **Magnitude**: Calculate the \( n \)th root of the magnitude. \[ r^{1/n} = \sqrt[n]{r} \]2. **Angles**: Determine the angles by dividing the main angle by \( n \), then apply successive increments of \( \frac{2\pi}{n} \) for additional roots. \[ \text{First angle: } \frac{\theta}{n}, \quad \text{Next angles: } \frac{\theta}{n} + k\frac{2\pi}{n} \] Here, \( k \) is an integer ranging from 0 to \( n-1 \). 3. **Write Roots in Polar Form**: Each root will be of the form: \[ r^{1/n} \left( \cos \left( \frac{\theta}{n} + k\frac{2\pi}{n} \right) + i \sin \left( \frac{\theta}{n} + k\frac{2\pi}{n} \right) \right) \]This method ensures all \( n \) distinct roots are discovered, each rotated evenly around the origin in the complex plane. Understanding these steps can help in solving various complex number problems efficiently.