A) 1
B) \[\cos \alpha +i\sin \alpha \]
C) \[\cos 3\alpha +i\sin 3\alpha \]
D) \[\cos 5\alpha +i\sin 5\alpha \]
Correct Answer: C
Solution :
\[{{(\cos \alpha +i\,\sin \alpha )}^{3/5}}={{[\cos 3\alpha +i\sin 3\alpha ]}^{1/5}}\] \[=\left[ \cos \frac{2n\pi +3\alpha }{5}+i\sin \frac{2n\pi +3\alpha }{5} \right]\] Required product \[=\left( \cos \frac{3\alpha }{5}+i\,\sin \frac{3\alpha }{5} \right)\] \[\left( \cos \frac{2\pi +3\alpha }{5}+i\sin \frac{2\pi +3\alpha }{5} \right)\] \[\times \left( \cos \frac{4\pi \times 3\alpha }{5}+i\sin \frac{4\pi +3\alpha }{5} \right)\left( \cos \frac{6\pi +3\alpha }{5}+i\sin \frac{6\pi +3\alpha }{5} \right)\] \[\times \left( \cos \frac{8\pi +3\alpha }{5}+i\,\,\sin \frac{8\pi +3\alpha }{5} \right)\] \[=\cos \left( \frac{3\alpha }{5}+\frac{2\pi }{5}+\frac{3\alpha }{5}+\frac{4\pi }{5}+\frac{3\alpha }{5}+\frac{6\pi }{5} \right.\] \[\left. +\frac{3\alpha }{5}+\frac{8\pi }{5}+\frac{3\alpha }{5} \right)\] \[=i\,\sin \left( \frac{3\alpha }{5}+\frac{2\pi }{5}+\frac{3\alpha }{5}+\frac{4\pi }{5}+\frac{3\alpha }{5}+\frac{6\pi }{5} \right.\] \[\left. +\frac{3\alpha }{5}+\frac{8\pi }{5}+\frac{3\alpha }{5} \right)\] \[=\cos \,(4\pi +3\alpha )+i\sin \,(4\pi +3\alpha )\] \[=\cos 3\alpha +i\,\sin 3\alpha \]
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