Phương pháp giải:

Lời giải chi tiết:

* $\Delta ABC$ :

$\begin{align}  A{{C}^{2}}=B{{A}^{2}}+B{{C}^{2}}-2BA.BC.\cos {{120}^{0}} \\  \,\,\,\,\,\,\,\,\,\,\,=4{{a}^{2}}+{{a}^{2}}-2.2a.a.\left( -\frac{1}{2} \right)=7{{a}^{2}}\Rightarrow AC=a\sqrt{7} \\ \end{align}$$\begin{align}  *\,\,\Delta DCB:\,\,B{{D}^{2}}=C{{D}^{2}}+C{{B}^{2}}-2CD.CD.\cos {{60}^{0}} \\  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=4{{a}^{2}}+{{a}^{2}}-2.2a.a.\frac{1}{2}=3{{a}^{2}} \\  \Rightarrow BD=a\sqrt{3}\Rightarrow SB=a\sqrt{6} \\ \end{align}$

* Vẽ $DE\bot AC$ có

${{S}_{\Delta ABC}}=\frac{1}{2}a.2a.\sin {{120}^{0}}=\frac{1}{2}DE.AC\Rightarrow DE=\frac{a\sqrt{3}}{\sqrt{7}}$.

* Vẽ giả tưởng $BH\bot \left( SAC \right)\Rightarrow \widehat{\left( SB;\left( SAC \right) \right)}=\widehat{BSH}$.

* $BH=d\left( B;\left( SAC \right) \right)=d\left( A;\left( SAC \right) \right)=DK$

$\frac{1}{D{{K}^{2}}}=\frac{1}{S{{D}^{2}}}+\frac{1}{D{{E}^{2}}}=\frac{1}{3{{a}^{2}}}+\frac{7}{3{{a}^{2}}}=\frac{8}{3{{a}^{2}}}\Rightarrow DK=\frac{a\sqrt{3}}{2\sqrt{2}}$.

* Tam giác vuông $SHB:\,\,\sin \widehat{BSH}=\frac{BH}{SB}=\frac{\sqrt{3}}{2\sqrt{2}}:a\sqrt{6}=\frac{1}{4}$

Chọn đáp án A.