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°)

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Identify the given complex number in polar form: \(9(\cos 30^\circ + i \sin 30^\circ)\). Here, the modulus \(r = 9\) and the argument \(\theta = 30^\circ\).

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, \(n=2\).

Calculate the modulus of the roots by taking the square root of \(r\): \(\sqrt{9} = 3\).

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 \(\frac{30^\circ + 360^\circ \times 1}{2}\).

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.

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주요 개념

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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.