In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 8(cos 210° + i sin 210°)

Verified step by step guidance

1. Identify the given complex number in polar form: \(8(\cos 210^\circ + i \sin 210^\circ)\), where the modulus \(r = 8\) and the argument \(\theta = 210^\circ\).
2. Recall that the cube roots of a complex number \(r(\cos \theta + i \sin \theta)\) are given by the formula: \(\sqrt[3]{r} \left( \cos \frac{\theta + 360^\circ k}{3} + i \sin \frac{\theta + 360^\circ k}{3} \right)\), where \(k = 0, 1, 2\).
3. Calculate the cube root of the modulus: \(\sqrt[3]{8} = 2\).
4. For each value of \(k = 0, 1, 2\), compute the argument of each root using \(\frac{210^\circ + 360^\circ k}{3}\).
5. Write each root in polar form as \(2 \left( \cos \theta_k + i \sin \theta_k \right)\), where \(\theta_k\) are the arguments found in the previous step.

Key Concepts

Complex Numbers in Polar Form

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 (angle). This form simplifies multiplication, division, and finding roots by working with magnitudes and angles separately.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, raising it to the nth power corresponds to raising the magnitude to the nth power and multiplying the angle by n. Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, adding multiples of 360°/n.

Finding nth Roots of Complex Numbers

To find all nth roots of a complex number, calculate the nth root of the magnitude and find n distinct angles by dividing the original angle by n and adding multiples of 360°/n. This yields n equally spaced roots on the complex plane in polar form.