The values of $\theta$, for which $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is real are
The values of $\theta$, for which $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is real are
- $\theta=n \pi+\frac{\pi}{3}$ for $n \in Z$
- $\theta=n \pi+\frac{\pi}{4}$ for $n \in Z$
- $\theta=n \pi+\frac{\pi}{2}$ for $n \in Z$
- $\theta=n \pi$ for $n \in Z$
Solution
$
\begin{gathered}
\frac{3+2 i \sin \theta}{1-2 i \sin \theta}=\frac{4}{1-2 i \sin \theta}-1 \\
=\frac{4(1+2 i \sin \theta)}{1+4 \sin ^2 \theta}-1
\end{gathered}
$
which is real.
$
\begin{array}{ll}
\therefore & 8 i \sin \theta=0 \Rightarrow \sin \theta=0 \\
\Rightarrow & \theta=n \pi, n \in Z
\end{array}
$
Asked in: AP EAMCET 2022 (07 Jul Shift 2)
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