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°)

Verified step by step guidance

1. Identify the given complex number in polar form: \(9(\cos 30^\circ + i \sin 30^\circ)\). Here, the modulus \(r = 9\) and the argument \(\theta = 30^\circ\).
2. Recall that to find the complex square roots of a number in polar form \(r(\cos \theta + i \sin \theta)\), we use the formula for the \(n\)th roots: \(\sqrt[n]{r} \left( \cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n} \right)\), where \(k = 0, 1, ..., n-1\). Since we want square roots, \(n=2\).
3. Calculate the modulus of the roots by taking the square root of \(r\): \(\sqrt{9} = 3\).
4. Calculate the arguments of the roots by dividing the original argument plus \(360^\circ k\) by 2 for \(k=0\) and \(k=1\): \(\frac{30^\circ + 360^\circ \times 0}{2}\) and \(\frac{30^\circ + 360^\circ \times 1}{2}\).
5. Write the two roots in polar form as \(3 \left( \cos \alpha + i \sin \alpha \right)\) where \(\alpha\) are the two 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 power n results in r^n (cos nθ + i sin nθ). Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, adding multiples of 360°/n for all roots.

Finding Complex Roots

To find the complex nth roots of a number, calculate the nth root of the magnitude and determine the arguments by dividing the original angle by n and adding k(360°/n) for k = 0, 1, ..., n-1. This yields all distinct roots evenly spaced around the circle.