选择章节

\((1)\)设\(\alpha \)为锐角， 若\(\cos \left( \alpha +\dfrac{\pi }{6} \right)=\dfrac{4}{5}\)，则\(\sin \alpha \)的值为_________．

\((2)\)在边长为\(2\)的正三角形\(ABC\)中，设\(\overrightarrow{BC}=2 \overrightarrow{BD}, \overrightarrow{CA}=3 \overrightarrow{CE} \)，则\(\overrightarrow{AD}\cdot \overrightarrow{BE}=\)_____________．

\((3)\)已知正四面体\(ABCD\)中，\(E\)是\(AB\)的中点，则异面直线\(CE\)与\(BD\)所成角的余弦值为__________．

\((4)\)若\(f\left( x \right)=A{\sin }\left( \omega +\varphi \right)+3(\omega > 0,\left| \varphi \right| < { }\!\!\pi\!\!{ })\)对任意实数\(t\)都有\(f\left( t+\dfrac{{ }\!\!\pi\!\!{ }}{3} \right)=f\left( -t+\dfrac{{ }\!\!\pi\!\!{ }}{3} \right).\)记\(g\left( x \right)=A{\cos }\left( \omega x+\varphi \right)-2\)，则\(g\left( \dfrac{{ }\!\!\pi\!\!{ }}{3} \right)=\)_______．

如图，在\(Rt\triangle AOB\)中，\(∠OAB=30^{\circ}\)，斜边\(AB=4\)，\(Rt\triangle AOC\)可以通过\(Rt\triangle AOB\)以直线\(AO\)为轴旋转得到，且二面角\(B-AO-C\)是直二面角\(.\)动点\(D\)在斜边\(AB\)上，则\(CD\)与平面\(AOB\)所成角的正切值的最大值_______．