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$A$ value of $\theta$, for which $\frac{2+3 i \sin \theta}{1-2 \sin \theta}, \mathrm{i}=\sqrt{-1}$ is purely imaginary, is
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The correct answer is:
$\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
$\mathrm{z}=\frac{2+3 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}=\frac{\left(2-6 \sin ^2 \theta\right)+\mathrm{i}(7 \sin \theta)}{1+4 \sin ^2 \theta}$
for $z$ to be purely imaginary $\operatorname{Re}(z)=0$
$\Rightarrow 6 \sin ^2 \theta=2$
$\Rightarrow \theta=\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
for $z$ to be purely imaginary $\operatorname{Re}(z)=0$
$\Rightarrow 6 \sin ^2 \theta=2$
$\Rightarrow \theta=\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
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