[Om Sharma]

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+\mathrm{i} \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-\mathrm{i} \cos \frac{2 \pi}{9}}\right)^{3}$ is [JEE (Main) 24 Jan 2023 S II]
(A) $\frac{1}{2}(1-\mathrm{i} \sqrt{3})$
(B) $-\frac{1}{2}(\sqrt{3}-\mathrm{i})$
(C) $-\frac{1}{2}(1-\mathrm{i} \sqrt{3})$
(D) $\frac{1}{2}(\sqrt{3}+\mathrm{i})$

Log in to download PDF

[Om Sharma]

Solution

Let $\sin \frac{2 \pi}{9}+\mathrm{i} \cos \frac{2 \pi}{9}=\mathrm{z} \Rightarrow\left(\frac{1+\mathrm{z}}{1+\bar{z}}\right)^{3}=\left(\frac{1+\mathrm{z}}{1+\frac{1}{\mathrm{x}}}\right)^{3}=\mathrm{z}^{3}$

\[ \left(\mathrm{i}\left(\cos \frac{2 \pi}{9}-\mathrm{i} \sin \frac{2 \pi}{9}\right)\right)^{3}=-\mathrm{i}\left(\cos \frac{2 \pi}{3}-\mathrm{i} \sin \frac{2 \pi}{3}\right)=-\mathrm{i}\left(-\frac{1}{2}-\mathrm{i} \frac{\sqrt{3}}{2}\right) \Rightarrow-\frac{1}{2}(\sqrt{3}-\mathrm{i}) \]

[Om Sharma]

Answer

B

[Author]

Posts