Example 6 Find the modulus of $2+2 e^{0.4 \pi i}+e^{1.2 \pi i}$.
(2009, Tsinghua University Independent Recruitment Examination)
Notice,
$$
\begin{array}{l}
12+2 \mathrm{e}^{0.4 \pi \mathrm{i}}+\mathrm{e}^{1.2 \pi \mathrm{i}} \\
=\sqrt{\left(2+2 \cos \frac{2 \pi}{5}+\cos \frac{6 \pi}{5}\right)^{2}+\left(2 \sin \frac{2 \pi}{5}+\sin \frac{6 \pi}{5}\right)^{2}} \\
=\sqrt{9+8 \cos \frac{2 \pi}{5}+4 \cos \frac{6 \pi}{5}+4 \cos \frac{4 \pi}{5}} \\
=\sqrt{9+8 \cos \frac{2 \pi}{5}-8 \cos \frac{\pi}{5}} \\
=\sqrt{9-16 \sin \frac{3 \pi}{10} \cdot \sin \frac{\pi}{10}} \\
=\sqrt{9-16 \cos 36^{\circ} \cdot \cos 72^{\circ}} \\
=\sqrt{9-\frac{16 \cos 36^{\circ} \cdot \cos 72^{\circ} \cdot \sin 36^{\circ}}{\sin 36^{\circ}}} \\
=\sqrt{9-\frac{4 \sin 144^{\circ}}{\sin 36^{\circ}}}=\sqrt{5} .
\end{array}
$$