Example 11 Let $x, y, z, A, B, C \in \mathbf{R}$, and
$$
A+B+C=\pi \text {. }
$$

Let $G_{n}=x^{n} \sin n A+y^{n} \sin n B+z^{n} \sin n C\left(n \in \mathbf{Z}_{+}\right)$.
If $G_{1}=G_{2}=0$, try to find the value of $G_{n}$.
【Analysis】From $G_{n}$, the $n$-fold angle in $G_{n}$ leads to the power formula of complex numbers. We can set
$$
\begin{array}{l}
z_{1}=x(\cos A+\mathrm{i} \sin A), \\
z_{2}=y(\cos B+\mathrm{i} \sin B), \\
z_{3}=z(\cos C+\mathrm{i} \sin C) .
\end{array}
$$

Consider the sequence $H_{n}=z_{1}^{n}+z_{2}^{n}+z_{3}^{n}$. Then
$$
G_{n}=\operatorname{Im} H_{n} \text {. }
$$

Given $G_{1}=G_{2}=0$, we only need to establish the recurrence relation for $H_{n}$.
Since $H_{n}$ is a recurrence sequence with characteristic roots $z_{1}, z_{2}, z_{3}$, we have
$$
\begin{array}{l}
z_{1}+z_{2}+z_{3}=H_{1} \in \mathbf{R}, \\
z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=H_{2} \in \mathbf{R}, \\
\text { so } z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=\frac{1}{2}\left(H_{1}^{2}-H_{2}\right) \in \mathbf{R}, \\
z_{1} z_{2} z_{3}=x y z \mathrm{e}^{\mathrm{i}(A+B+C)}=-x y z \in \mathbf{R} . \\
\text { Therefore, } H_{n+3}=H_{1} H_{n+2}-\frac{1}{2}\left(H_{1}^{2}-H_{2}\right) H_{n+1}-x y z H_{n}
\end{array}
$$

is a recurrence relation with real coefficients.
Thus, by $H_{0}, H_{1}, H_{2} \in \mathbf{R}$ and induction, we know $H_{n} \in \mathbf{R}$.
Therefore, $G_{n}=\operatorname{Im} H_{n}=0$.