In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of −1

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Recognize that finding the complex cube roots of \(-1\) means solving the equation \(z^3 = -1\). We want to find all complex numbers \(z\) such that when raised to the third power, they equal \(-1\).

Express \(-1\) in its polar (trigonometric) form. Since \(-1\) lies on the real axis at an angle of \(\pi\) radians (180 degrees) from the positive real axis, we write \(-1 = 1 \cdot (\cos \pi + i \sin \pi)\).

Use De Moivre's Theorem to find the cube roots. The general formula for the \(n\)-th roots of a complex number \(r(\cos \theta + i \sin \theta)\) is given by: \[ z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right) \] where \(k = 0, 1, ..., n-1\). Here, \(n=3\), \(r=1\), and \(\theta=\pi\).

Calculate each root by substituting \(k=0, 1, 2\) into the formula: \[ z_k = \cos \left( \frac{\pi + 2k\pi}{3} \right) + i \sin \left( \frac{\pi + 2k\pi}{3} \right) \] This will give three distinct roots on the complex plane.

Convert each root from polar form to rectangular form by evaluating the cosine and sine values for each angle. This will give the roots in the form \(a + bi\), where \(a = \cos(\text{angle})\) and \(b = \sin(\text{angle})\).

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주요 개념

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Complex Roots of Unity

Complex roots of unity are solutions to the equation z^n = 1, where n is a positive integer. These roots are evenly spaced points on the unit circle in the complex plane, each separated by an angle of 2π/n radians. Understanding these roots helps in finding roots of other complex numbers by relating them to roots of unity.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Polar form is useful for finding roots and powers using De Moivre's theorem, while rectangular form is often preferred for final answers. Converting between these forms is essential for solving and expressing complex roots.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). This theorem is fundamental for finding nth roots of complex numbers by taking the nth root of the magnitude and dividing the angle by n, then considering all possible angles separated by 2π/n.