Understanding De Moivre's Theorem
De Moivre's Theorem is a powerful tool in complex analysis that connects complex numbers with trigonometry. It states that for any real number $\theta$ and any integer $n$:
$$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$$
This theorem is exceptionally useful for finding powers and roots of complex numbers represented in polar form.
Finding the Fourth Roots of a Complex Number
We are tasked to find the fourth roots of the complex number $z = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$.
Step 1: Convert to Polar Form
First, we find the modulus $r$ and the argument $\theta$ of $z$.
- $r = \sqrt{(-\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1$
- Since the real part is negative and the imaginary part is positive, the angle $\theta$ is in the second quadrant.
- $\theta = \operatorname{atan2}(\frac{\sqrt{3}}{2}, -\frac{1}{2}) = \frac{2\pi}{3}$
So, $z = 1(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3})$.
Step 2: Applying the Generalized De Moivre's Theorem
To find the $n$-th roots, we use the formula: $$z_k = r^{1/n} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right]$$ where $k = 0, 1, 2, 3$ for fourth roots ($n=4$).
Step 3: Calculate the Four Roots
Substitute $r=1, \theta = \frac{2\pi}{3}, n=4$:
- For k=0: $$z_0 = \cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} + \frac{1}{2}i$$
- For k=1: $$z_1 = \cos(\frac{2\pi/3 + 2\pi}{4}) + i\sin(\dots) = \cos(\frac{8\pi}{12}) + i\sin(\frac{8\pi}{12}) = \cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$$
- For k=2: $$z_2 = \cos(\frac{2\pi/3 + 4\pi}{4}) + i\sin(\dots) = \cos(\frac{14\pi}{12}) + i\sin(\frac{14\pi}{12}) = \cos(\frac{7\pi}{6}) + i\sin(\frac{7\pi}{6}) = -\frac{\sqrt{3}}{2} - \frac{1}{2}i$$
- For k=3: $$z_3 = \cos(\frac{2\pi/3 + 6\pi}{4}) + i\sin(\dots) = \cos(\frac{20\pi}{12}) + i\sin(\frac{20\pi}{12}) = \cos(\frac{5\pi}{3}) + i\sin(\frac{5\pi}{3}) = \frac{1}{2} - i\frac{\sqrt{3}}{2}$$
Summary
By expressing the complex number in polar coordinates and utilizing the periodicity of trigonometric functions, we can effectively split the rotation space into $n$ equal parts to extract all roots systematically.