Example 8. In the rectangular cuboid $A B C D-A_{1} B_{1} C \cdot D_{1}$ (Fig. 15), $A B=2, A A_{1}=A D=1$. Find the angle between line $A B$ and plane $A B_{1} C$.

Let the angle between $AB$ and the plane $AB_{1}C$ be $a$.
$$
\angle B A B_{1}=\angle B A C=\arccos \frac{2}{\sqrt{5}} \cdot
$$

By the corollary of formula 1,
$$
\begin{aligned}
\cos \alpha=\frac{\cos \angle B A B_{1}}{\cos \frac{\angle B_{1} A C}{2}} & =\frac{\frac{2}{\sqrt{5}}}{\frac{3}{\sqrt{10}}} \\
& =\frac{2 \sqrt{2}}{3} .
\end{aligned}
$$