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uysha [10]
2 years ago
8

Find the cube roots of 8(cos 216° + i sin 216°).

Mathematics
1 answer:
Tomtit [17]2 years ago
5 0
z^3=8(\cos216^\circ+i\sin216^\circ)
z^3=2^3(\cos(6^3)^\circ+i\sin(6^3)^\circ)
\implies z=8^{1/3}\left(\cos\left(\dfrac{216+360k}3\right)^\circ+i\sin\left(\dfrac{216+360k}3\right)^\circ\right)

where k=0,1,2. So the third roots are

z=\begin{cases}2(\cos72^\circ+i\sin72^\circ)\\2(\cos192^\circ+i\sin192^\circ)\\2(\cos312^\circ+i\sin312^\circ)\end{cases}
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