Let us find the six roots of the equation z 6 = − 1 z^6=-1 z 6 = − 1 using de Moivre's Theorem for fractional power:
[ r ( cos θ + i sin θ ) ] 1 n = r n ( cos θ + 2 π k n + i sin θ + 2 π k n ) [r(\cos\theta+i\sin\theta)]^{\frac{1}{n}}=\sqrt[n]{r}(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}) [ r ( cos θ + i sin θ ) ] n 1 = n r ( cos n θ + 2 πk + i sin n θ + 2 πk ) for k ∈ { 0 , 1 , . . . , n − 1 } . k\in\{0,1,...,n-1\}. k ∈ { 0 , 1 , ... , n − 1 } .
In trigonometric form − 1 = cos π + i sin π -1=\cos\pi+i\sin\pi − 1 = cos π + i sin π , that is r = 1 , θ = π . r=1, \theta=\pi. r = 1 , θ = π . Consequently, the formula
z k = cos π + 2 π k 6 + i sin π + 2 π k 6 = e π + 2 π k 6 i z_k=\cos\frac{\pi+2\pi k}{6}+i\sin\frac{\pi+2\pi k}{6}=e^{\frac{\pi+2\pi k}{6}i} z k = cos 6 π + 2 πk + i sin 6 π + 2 πk = e 6 π + 2 πk i for k ∈ { 0 , 1 , 2 , 3 , 4 , 5 } k\in\{0,1,2,3,4,5\} k ∈ { 0 , 1 , 2 , 3 , 4 , 5 } gives all six different roots.
Therefore, we have:
z 0 = e π 6 i z_0=e^{\frac{\pi}{6}i} z 0 = e 6 π i , z 1 = e 3 π 6 i = e π 2 i z_1=e^{\frac{3\pi}{6}i}=e^{\frac{\pi}{2}i} z 1 = e 6 3 π i = e 2 π i , z 2 = e 5 π 6 i z_2=e^{\frac{5\pi}{6}i} z 2 = e 6 5 π i , z 3 = e 7 π 6 i z_3=e^{\frac{7\pi}{6}i} z 3 = e 6 7 π i , z 4 = e 9 π 6 i = e 3 π 2 i z_4=e^{\frac{9\pi}{6}i}=e^{\frac{3\pi}{2}i} z 4 = e 6 9 π i = e 2 3 π i , z 5 = e 11 π 6 i . z_5=e^{\frac{11\pi}{6}i}. z 5 = e 6 11 π i .
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