SAT1000 - P322

Given the function $f(x)=\sin^2 \dfrac{\omega x}{2}+\dfrac{1}{2} \sin \omega x - \dfrac{1}{2}\ (\omega>0)$, $x \in \mathbb R$.

If $f(x)=0$ has no roots for $x \in (\pi, 2\pi)$, then find the range of $\omega$.

$A.\ (0,\dfrac{1}{8}]$

$B.\ (0,\dfrac{1}{4}] \cup [\dfrac{5}{8},1]$

$C.\ (0,\dfrac{5}{8}]$

$D.\ (0,\dfrac{1}{8}] \cup [\dfrac{1}{4},\dfrac{5}{8}]$

Have a look at my problem set: SAT 1000 problems