Ta có: \(\left\{\begin{array}{l}(S B C) \cap(A B C)=B C \\ \text{Trong}(A B C), O I \perp B C \Rightarrow[S, B C, O]=S I O \\ \text{Trong}(S B C), S I \perp B C\end{array}\right.\)

Ta có: \(O I=\frac{1}{3} \cdot \frac{a \sqrt{3}}{2}=\frac{a \sqrt{3}}{6}, S O=\sqrt{S B^{2}-O B^{2}}=\sqrt{(2 a)^{2}-\left(\frac{a \sqrt{3}}{3}\right)^{2}}=\frac{\sqrt{33}}{3} a\)

Xét \(\triangle S O I\) vuông tại \(O: \tan S I O=\frac{S O}{O I}=\frac{\frac{\sqrt{33} a}{3}}{\frac{a \sqrt{3}}{6}}=2 \sqrt{11} \Rightarrow S I O \approx 81,4^{0}\)

\(\Rightarrow[S, B C, O] \approx 81,4^{0}\)