3. In a circle, the central angles corresponding to chords of lengths $2$, $3$, and $4$ are $\alpha, \beta$, and $\alpha+\beta$, respectively, where $\alpha+\beta<\pi, \cos \alpha=$ $\qquad$
3. $\frac{17}{32}$.
$AB, BC$ are two chords of $\odot O$, the radian measures of $\overparen{AB}$ and $\overparen{BC}$ are $\alpha$ and $\beta$ respectively, so the radian measure of $\overparen{AC}$ is $\alpha+\beta$,
$$
\begin{array}{l}
AB=2, BC=3, AC=4 . \text{ In } \triangle ABC, \angle ACB=\frac{\alpha}{2}, \cos \angle ACB=\cos \frac{\alpha}{2}= \\
\frac{AC^{2}+BC^{2}-AB^{2}}{2 \cdot AC \cdot BC}=\frac{7}{8}, \cos \alpha=2 \cos ^{2} \frac{\alpha}{2}-1=\frac{17}{32} .
\end{array}
$$