Example 1 Find the maximum value of the function $y=\sin \alpha+2 \sin \beta+3 \sin \gamma+4 \sin \theta$, where $\alpha+\beta+$ $\gamma+\theta=\pi$.
Solving: In equation (2), taking $x=1, y=2, z=3, w=4$, we get
$$\sin \alpha+2 \sin \beta+3 \sin \gamma+4 \sin \theta \leqslant \sqrt{\frac{385}{6}}$$

That is, $y$ has a maximum value of $\sqrt{\frac{385}{6}}$.