In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 1

Verified step by step guidance

1. Identify the problem as finding the sixth roots of the complex number 1, which can be expressed as a complex number in polar form: $1 = e^{i \cdot 0}$.
2. Use De Moivre's Theorem, which states that the $n$th roots of a complex number $r e^{i\theta}$ are given by $r^{1/n} e^{i(\theta + 2k\pi)/n}$ for $k = 0, 1, 2, \ldots, n-1$.
3. Since the magnitude $r$ of 1 is 1, the magnitude of each root will also be 1. Therefore, the sixth roots are $e^{i(0 + 2k\pi)/6}$ for $k = 0, 1, 2, 3, 4, 5$.
4. Calculate the angles for each root: $\theta_k = \frac{2k\pi}{6}$, which simplifies to $\theta_k = \frac{k\pi}{3}$ for $k = 0, 1, 2, 3, 4, 5$.
5. Express each root in rectangular form using $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, resulting in the roots: $\cos(\frac{k\pi}{3}) + i\sin(\frac{k\pi}{3})$ for each $k$.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. They are essential in various fields of mathematics and engineering, particularly in solving equations that do not have real solutions. Understanding how to manipulate and represent complex numbers is crucial for finding roots of complex equations.

Roots of Unity

Roots of unity are complex numbers that satisfy the equation z^n = 1, where 'n' is a positive integer. The nth roots of unity are evenly spaced around the unit circle in the complex plane, and they can be expressed in polar form as e^(2πik/n) for k = 0, 1, ..., n-1. This concept is vital for solving problems involving complex roots, as it provides a systematic way to find all roots.

Rectangular Form

Rectangular form refers to the representation of complex numbers as a + bi, where 'a' and 'b' are real numbers. Converting complex numbers from polar form (r(cos θ + i sin θ)) to rectangular form is often necessary for practical applications, such as addition and subtraction of complex numbers. Understanding how to perform this conversion is essential when expressing roots in the required format.