We'll use the notation [tex]\textrm{Arcsin } x[/tex] for the principal value and [tex]\arcsin x[/tex] for all the values:
[tex] \arcsin x = \textrm{Arcsin } x + 2\pi k \textrm{ or } (\pi - \textrm{Arcsin }x) + 2\pi k \quad \textrm{ integer } k[/tex]
Part I
[tex] \sin(\theta/2) = \frac 1 2 [/tex]
[tex] \theta/2 = \arcsin \frac 1 2 [/tex]
Part II. In our range we can write
[tex] \sin(\theta/2) = \sin(\frac \pi 6)[/tex]
[tex]\theta/2 = \frac \pi 6 \textrm{ or } \theta /2 = \pi - \frac \pi 6 = \frac{5\pi}{6}[/tex]
Part III.
[tex]\theta = \frac \pi 3 \textrm{ or } \theta = \frac{5\pi}{3}[/tex]
Part IV.
[tex] \theta/2 = \frac \pi 6 + 2\pi k \textrm{ or } \theta = \frac{5 \pi}{6} + 2\pi k \quad \textrm{integer } k[/tex]
[tex] \theta = \frac \pi 3 + 4 \pi k \textrm{ or } \theta = \frac{5 \pi}{3} + 4\pi k \quad \textrm{integer } k[/tex]
