Bepaal $$$\sqrt[3]{i}$$$

Bepaal $$$\sqrt[3]{i}$$$.

De polaire vorm van $$$i$$$ is $$$\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}$$$ (voor de stappen, zie rekenmachine voor polaire vorm).

Volgens de formule van De Moivre worden alle $$$n$$$-de wortels van een complex getal $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$ gegeven door $$$r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right)$$$, $$$k=\overline{0..n-1}$$$.

We hebben dat $$$r = 1$$$, $$$\theta = \frac{\pi}{2}$$$ en $$$n = 3$$$.

• $$$k = 0$$$: $$$\sqrt[3]{1} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{3} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{3} \right)}\right) = \cos{\left(\frac{\pi}{6} \right)} + i \sin{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$$
• $$$k = 1$$$: $$$\sqrt[3]{1} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{3} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{3} \right)}\right) = \cos{\left(\frac{5 \pi}{6} \right)} + i \sin{\left(\frac{5 \pi}{6} \right)} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$$
• $$$k = 2$$$: $$$\sqrt[3]{1} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{3} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{3} \right)}\right) = \cos{\left(\frac{3 \pi}{2} \right)} + i \sin{\left(\frac{3 \pi}{2} \right)} = - i$$$

$$$\sqrt[3]{i} = \frac{\sqrt{3}}{2} + \frac{i}{2}\approx 0.866025403784439 + 0.5 i$$$A

$$$\sqrt[3]{i} = - \frac{\sqrt{3}}{2} + \frac{i}{2}\approx -0.866025403784439 + 0.5 i$$$A

$$$\sqrt[3]{i} = - i$$$A