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. Recognize that the problem asks for the sixth roots of 1, which means we want to find all complex numbers \( z \) such that \( z^6 = 1 \).
2. Express 1 in its complex polar form. Since \(1\) lies on the positive real axis, its magnitude is 1 and its argument (angle) is 0. So, \(1 = 1 \left( \cos 0 + i \sin 0 \right)\).
3. Use De Moivre's Theorem to find the sixth roots. The general formula for the \(n\)th roots of a complex number \(r(\cos \theta + i \sin \theta)\) is given by: \[ z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2\pi k}{n} \right) + i \sin \left( \frac{\theta + 2\pi k}{n} \right) \right) \quad \text{for} \quad k = 0, 1, 2, ..., n-1 \] Here, \(n=6\), \(r=1\), and \(\theta=0\).
4. Calculate each root by substituting \(k = 0, 1, 2, 3, 4, 5\) into the formula to find the six distinct roots. Each root will have the form: \[ z_k = \cos \left( \frac{2\pi k}{6} \right) + i \sin \left( \frac{2\pi k}{6} \right) \]
5. Convert each root from polar to rectangular form by evaluating the cosine and sine values for each \(k\). If necessary, round the real and imaginary parts to the nearest tenth to express the roots in rectangular form \(a + bi\).

Key Concepts

Complex Roots of Unity

Complex roots of unity are the solutions to the equation z^n = 1, where n is a positive integer. These roots are evenly spaced points on the unit circle in the complex plane, each separated by an angle of 2π/n radians. For the sixth roots of 1, there are six such roots.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument. To write roots in rectangular form, convert using x = r cos θ and y = r sin θ, resulting in x + yi. This conversion is essential for expressing roots as requested.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It is used to find the nth roots by taking the nth root of r and dividing the angle θ by n, generating all distinct roots by adding multiples of 2π/n.