In Exercises 30–31, find all the complex roots. Write roots in polar form with θ in degrees.The complex cube roots of 125(cos 165° + i sin 165°)

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 their roots and performing operations in the complex plane.

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 coordinates.

Polar coordinates represent a point in the complex plane using a distance from the origin (the modulus) and an angle from the positive x-axis (the argument). In the context of complex numbers, converting to polar form is crucial for operations like multiplication, division, and finding roots. The angle θ is typically measured in degrees or radians, and understanding how to manipulate these coordinates is key to solving problems involving complex roots.