Question

PLEASE ANSWER ASAP!!!!

Answer 1

We'll use the notation $\textrm{Arcsin } x$ for the principal value and $\arcsin x$ for all the values:

$\arcsin x = \textrm{Arcsin } x + 2\pi k \textrm{ or } (\pi - \textrm{Arcsin }x) + 2\pi k \quad \textrm{ integer } k$

Part I

$\sin(\theta/2) = \frac 1 2$

$\theta/2 = \arcsin \frac 1 2$

Part II. In our range we can write

$\sin(\theta/2) = \sin(\frac \pi 6)$

$\theta/2 = \frac \pi 6 \textrm{ or } \theta /2 = \pi - \frac \pi 6 = \frac{5\pi}{6}$

Part III.

$\theta = \frac \pi 3 \textrm{ or } \theta = \frac{5\pi}{3}$

Part IV.

$\theta/2 = \frac \pi 6 + 2\pi k \textrm{ or } \theta = \frac{5 \pi}{6} + 2\pi k \quad \textrm{integer } k$

$\theta = \frac \pi 3 + 4 \pi k \textrm{ or } \theta = \frac{5 \pi}{3} + 4\pi k \quad \textrm{integer } k$