In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)

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 trigonometry, complex numbers can also be represented in polar form as r(cos θ + i sin θ), where 'r' is the magnitude and 'θ' is the angle. Understanding this representation is crucial for finding roots of complex numbers.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude and angle, using the formula r(cos θ + i sin θ). The magnitude 'r' is calculated as the square root of the sum of the squares of the real and imaginary parts, while 'θ' is the angle formed with the positive real axis. This form is particularly useful for operations like multiplication, division, and finding roots of complex numbers.

Finding Roots of Complex Numbers

To find the roots of a complex number in polar form, we use De Moivre's Theorem, which states that for a complex number 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))) for k = 0, 1, ..., n-1. This allows us to determine all distinct roots by varying 'k', which is essential for solving the given problem.