In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 27(cos 306° + i sin 306°)

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. In trigonometric form, a complex number can be represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument. Understanding complex numbers is essential for finding roots and performing operations in the complex plane.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ), the nth roots can be found using the formula r^(1/n)(cos(θ/n + k(360°/n)) + i sin(θ/n + k(360°/n))), where k is an integer from 0 to n-1. This theorem simplifies the process of finding roots of complex numbers by allowing us to work directly with their polar representations.

Polar Coordinates

Polar coordinates represent points in the complex plane using a distance from the origin (r) and an angle (θ) from the positive x-axis. This system is particularly useful for complex numbers, as it allows for easier multiplication, division, and finding roots. Converting between rectangular and polar forms is a key skill in trigonometry and complex analysis.