In which quadrant is terminal side of angle 2$\theta$ and $\theta / 2$? How do I find the quadrants of 2$\theta$ and $\theta /2$ if
sin $\theta$ = 4/5 is in 2nd quadrant.

The phrase "$\sin \theta = 4/5$ is in the second quadrant" is malformed.  It is a splice of two ideas: "$\sin \theta = 4/5$" and "$\theta$ is in the second quadrant".  It is indicative of confusion to say that the side length ratio $\frac{4}{5}$ is in any quadrant.
Since $\theta$ is in the second quadrant, $\dfrac{\pi}{2} < \theta < \pi$.  Dividing through by $2$, we find $\dfrac{\pi}{4} < \dfrac{\theta}{2} < \dfrac{\pi}{2}$, so $\dfrac{\theta}{2}$ is in the first quadrant.
We can also double this equation, $\pi < 2 \theta < 2 \pi$, so $2 \theta$ is in the third or fourth quadrants.  Sine is the same sign in both quadrants, so $\sin 2 \theta$ is of no use in discriminating between the two possibilities.  Cosine, however, is positive in the fourth quadrant and negative in the third quadrant, so the sign of cosine will resolve the question.  $\cos 2 \theta = 1 - 2 \sin^2(\theta) = 1- 2 (\frac{4}{5})^2 = 1 - \frac{32}{25} < 0$ and we find that $2\theta$ is in the third quadrant.
