3. Let the function $f(x)=A \sin (\omega x+\varphi)(A \neq 0$, $\left.\omega>0,0<\varphi<\frac{\pi}{2}\right)$. If $f\left(\frac{5 \pi}{6}\right)+f(0)=0$, then the minimum value of $\omega$ is . $\qquad$
3. $\frac{6}{5}$.
$$
\begin{array}{l}
\text { Given } f(0)+f\left(\frac{5 \pi}{6}\right) \\
=A \sin \varphi+A \sin \left(\frac{5 \pi \omega}{6}+\varphi\right) \\
=2 A \sin \left(\frac{5 \pi \omega}{12}+\varphi\right) \cdot \cos \frac{5 \pi \omega}{12}=0 \\
\Rightarrow \frac{5 \pi \omega}{12}=\frac{\pi}{2}+2 k \pi \\
\Leftrightarrow \omega=\frac{6}{5}+\frac{24 k}{5}(k \in \mathbf{Z}) .
\end{array}
$$

From $\omega>0$, we know $\omega_{\text {min }}=\frac{6}{5}$.