Problem 24 - Contemporary Precalculus

Question

Find the nth roots in polar form. $$1 ; \quad n=7$$

Step-by-Step Solution

Answer

Answer: The 7th roots of 1 in polar form are: $$\cos[\frac{0}{7}] + i\sin[\frac{0}{7}],\quad \cos[\frac{2\pi}{7}] + i\sin[\frac{2\pi}{7}],\quad \cos[\frac{4\pi}{7}] + i\sin[\frac{4\pi}{7}],\quad \cos[\frac{6\pi}{7}] + i\sin[\frac{6\pi}{7}],\quad \cos[\frac{8\pi}{7}] + i\sin[\frac{8\pi}{7}],\quad \cos[\frac{10\pi}{7}] + i\sin[\frac{10\pi}{7}],\quad \cos[\frac{12\pi}{7}] + i\sin[\frac{12\pi}{7}]$$

Step 1: Convert the complex number to polar form

First, we need to find the polar form of the complex number 1. The polar form of a complex number is given by the equation: $$z = r(\cos[\theta] + i\sin{[\theta]})$$ We start by converting the complex number 1 to its polar form. In our case, the complex number is 1, which lies on the real axis at angle θ = 0. Therefore, the polar form of 1 is: $$z = 1(\cos[0] + i\sin[0])$$

Step 2: Apply De Moivre's theorem to find the 7th roots

Now that we have the polar form of the complex number 1, we can use De Moivre's theorem to find the 7th roots. De Moivre's theorem tells us that: $$(r(\cos[\theta]+i\sin{[\theta]}))^{\frac{1}{n}} = r^{\frac{1}{n}}\left(\cos\left[\frac{\theta+2k\pi}{n}\right] + i \sin\left[\frac{\theta+2k\pi}{n}\right]\right),\quad k = 0,1,2,...,n-1$$ Using De Moivre's theorem, we'll find each of the 7th roots by plugging in our values and calculating the polar form for each root. In our case, r = 1 and n = 7.

Step 3: Calculate each 7th root

We calculate each 7th root for k = 0 to 6: k = 0: $$1^{\frac{1}{7}}(\cos[\frac{0+2(0)\pi}{7}] + i\sin[\frac{0+2(0)\pi}{7}]) = \cos[\frac{0}{7}] + i\sin[\frac{0}{7}]$$ k = 1: $$1^{\frac{1}{7}}(\cos[\frac{0+2(1)\pi}{7}] + i\sin[\frac{0+2(1)\pi}{7}]) = \cos[\frac{2\pi}{7}] + i\sin[\frac{2\pi}{7}]$$ k = 2: $$1^{\frac{1}{7}}(\cos[\frac{0+2(2)\pi}{7}] + i\sin[\frac{0+2(2)\pi}{7}]) = \cos[\frac{4\pi}{7}] + i\sin[\frac{4\pi}{7}]$$ k = 3: $$1^{\frac{1}{7}}(\cos[\frac{0+2(3)\pi}{7}] + i\sin[\frac{0+2(3)\pi}{7}]) = \cos[\frac{6\pi}{7}] + i\sin[\frac{6\pi}{7}]$$ k = 4: $$1^{\frac{1}{7}}(\cos[\frac{0+2(4)\pi}{7}] + i\sin[\frac{0+2(4)\pi}{7}]) = \cos[\frac{8\pi}{7}] + i\sin[\frac{8\pi}{7}]$$ k = 5: $$1^{\frac{1}{7}}(\cos[\frac{0+2(5)\pi}{7}] + i\sin[\frac{0+2(5)\pi}{7}]) = \cos[\frac{10\pi}{7}] + i\sin[\frac{10\pi}{7}]$$ k = 6: $$1^{\frac{1}{7}}(\cos[\frac{0+2(6)\pi}{7}] + i\sin[\frac{0+2(6)\pi}{7}]) = \cos[\frac{12\pi}{7}] + i\sin[\frac{12\pi}{7}]$$ So, the 7th roots of 1 in polar form are: $$\cos[\frac{0}{7}] + i\sin[\frac{0}{7}],\quad \cos[\frac{2\pi}{7}] + i\sin[\frac{2\pi}{7}],\quad \cos[\frac{4\pi}{7}] + i\sin[\frac{4\pi}{7}],\quad \cos[\frac{6\pi}{7}] + i\sin[\frac{6\pi}{7}],\quad \cos[\frac{8\pi}{7}] + i\sin[\frac{8\pi}{7}],\quad \cos[\frac{10\pi}{7}] + i\sin[\frac{10\pi}{7}],\quad \cos[\frac{12\pi}{7}] + i\sin[\frac{12\pi}{7}]$$

Key Concepts

Polar Form

The polar form of a complex number provides a different perspective compared to the usual rectangular form (a + bi). It emphasizes the number's magnitude and angle with respect to the positive real axis. This interpretation is particularly useful in multiplication and division, as well as finding roots of complex numbers.

In polar form, a complex number is expressed as:

• Magnitude \( r \): The distance of the complex number from the origin in the complex plane. It is given by \( r = \sqrt{a^2 + b^2} \).
• Argument \( \theta \): The angle formed with the positive x-axis, found using \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \).

The polar form is thus written as:\[z = r(\cos\theta + i\sin\theta)\]This form makes it straightforward to apply trigonometric identities when performing complex number operations. For example, the complex number 1 is simply \( z = 1(\cos 0 + i\sin 0) \) in polar form, since it lies on the positive real axis.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool when dealing with exponential properties of complex numbers. It states:\[(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\]This theorem simplifies the process of raising complex numbers to powers or extracting roots. By transforming the number into polar form, the calculations are reduced to basic arithmetic operations on the modulus and angles.

When finding the nth root of a complex number, De Moivre’s Theorem gives a formula:\[(r(\cos\theta + i\sin\theta))^{\frac{1}{n}} = r^{\frac{1}{n}} \left(\cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right)\]where \( k = 0, 1, 2, ..., n-1 \). This formula generates all n roots of the complex number. For the exercise of finding the 7th roots of 1:

• Magnitude \( r \) is 1, so all roots have the same size.
• The angles \( \frac{\theta + 2k\pi}{7} \) distribute the roots evenly about the circle, forming a regular heptagon.

Complex Numbers

Complex numbers expand the real number system to include solutions to equations that are insoluble within just the real numbers. Each complex number is represented as \( a + bi \), where \( i \) is the imaginary unit such that \( i^2 = -1 \).

These numbers can be visualized in a plane where:

• Real part \( a \) corresponds to the horizontal axis (x-axis).
• Imaginary part \( b \) corresponds to the vertical axis (y-axis).

Understanding complex numbers includes grasping the ideas of addition, subtraction, multiplication, and division in this form.

To convert to polar form, what we essentially do is represent this space as rotations (angles) and stretches (magnitudes). This can lend great insight, especially in engineering fields, where complex numbers describe wave functions, electrical currents, and more. The polar or exponential forms can clarify behaviors and simplify computations that would otherwise involve intricate algebraic expressions.