若 $\sin \alpha+\sqrt{3} \cos \alpha=\frac{1}{2}$, 则 $(\quad)$
A
$\cos \left(\alpha+\frac{5 \pi}{6}\right)=\frac{1}{4}$
B
$3 \tan ^2 \alpha+8 \sqrt{3} \tan \alpha=-11$
C
$\sin \left(\alpha+\frac{4 \pi}{3}\right)=-\frac{1}{4}$
D
$3 \tan ^2 \alpha+8 \sqrt{3} \tan \alpha=-12$
E
F