5.4. When rotated around the origin, point $A(6 ; 8)$ moves to point $A_{1}(8 ; 6)$. Find the cosine of the rotation angle.
5.4. Using formula (5.21), we find

$$
\cos \angle A O A_{1}=\frac{\overline{O A} \cdot \overline{O A_{1}}}{|\overline{O A}| \overline{O A_{1}}|}=\frac{6 \cdot 8+8 \cdot 6}{\sqrt{36+64} \cdot \sqrt{64+36}}=\frac{24}{25}
$$