An object is in equilibrium under four concurrent forces in the directions shown in figure. Find the magnitudes of ${\overrightarrow{F}}_{1}and{\overrightarrow{F}}_2.$

The object is in equilibrium. Hence,
(i) $\Sigma F_x=0$
$\therefore 8+4\cos {60}^{\circ }-F_2\cos {30}^{\circ }=0$
or $8+2-F_2\frac{\sqrt{3}}{2}=0$
or $F_2=\frac{20}{\sqrt{3}}N$
(ii) $\Sigma F_y=0$
$\therefore F_1+4\sin {60}^{\circ }-F_2\sin {30}^{\circ }=0$
or $F_1+\frac{4\sqrt{3}}{2}-\frac{F_2}{2}=0$
or $F_1=\frac{F_2}{2}-2\sqrt{3}=\frac{10}{\sqrt{3}}-2\sqrt{3}$
or $F_1=\frac{4}{\sqrt{3}}N$