If $\sin \mathrm{\theta },1,\cos 2\mathrm{\theta }$are in G.P., then the general values of $\mathrm{\theta }$

$\begin{array}{l}{1}^2=\mathrm{sin\theta cos}2\mathrm{\theta }\text{ or }1-\mathrm{sin\theta }\left(1-2{\sin }^2\mathrm{\theta }\right)=0\\ \Rightarrow 2 {\sin }^3\mathrm{\theta }-\mathrm{sin\theta }+1=0\Rightarrow \mathrm{sin\theta }=-1\end{array}$
The other factor gives imaginary roots
$\begin{array}{l}\Rightarrow \mathrm{\theta }=\mathrm{n\pi }+(-1{)}^{\mathrm{n}}\left(-\frac{\mathrm{\pi }}{2}\right)\\ =\mathrm{n\pi }-(-1{)}^{\mathrm{n}}\frac{\mathrm{\pi }}{2}=\mathrm{n\pi }+(-1{)}^{\mathrm{n}-1}\frac{\mathrm{\pi }}{2}\end{array}$