2. The maximum value of the function $f(x)=7 \sin x+\sin 2 x$ is $\qquad$ .
2. $\frac{15 \sqrt{15}}{8}$.

From the condition, we know
$$
\begin{array}{l}
f^{\prime}(x)=7 \cos x+2 \cos 2 x \\
=4 \cos ^{2} x+7 \cos x-2 \\
=(4 \cos x-1)(\cos x+2) .
\end{array}
$$

Let $\cos \varphi=\frac{1}{4}\left(0<\varphi<\frac{\pi}{2}\right)$. Then
$$
\sin \varphi=\frac{\sqrt{15}}{4} \text {. }
$$

It is easy to see that $f(x)$ is monotonically increasing in the interval $(-\varphi+2 k \pi, \varphi+2 k \pi)$ $(k \in \mathbf{Z})$, and monotonically decreasing in the interval $(\varphi+2 k \pi, 2 \pi-\varphi+2 k \pi)(k \in \mathbf{Z})$. 
Therefore, the maximum value of $f(x)$ is
$$
f(\varphi+2 k \pi)=\frac{15 \sqrt{15}}{8} .
$$