3.098. $\sin \left(2 \alpha-\frac{3}{2} \pi\right)+\cos \left(2 \alpha-\frac{8}{3} \pi\right)+\cos \left(\frac{2}{3} \pi+2 \alpha\right)$.
## Solution.

Let

$$
\begin{aligned}
& X=\sin \left(2 \alpha-\frac{3}{2} \pi\right)+\cos \left(2 \alpha-\frac{8}{3} \pi\right)+\cos \left(\frac{2}{3} \pi+2 \alpha\right)= \\
& =-\sin \left(\frac{3}{2} \pi-2 \alpha\right)+\cos \left(\frac{8}{3} \pi-2 \alpha\right)+\cos \left(\frac{2}{3} \pi+2 \alpha\right) \\
& -\sin \left(\frac{3}{2} \pi-2 \alpha\right)=\cos 2 \alpha \\
& \cos \left(\frac{8}{3} \pi-2 \alpha\right)=\cos \left(\frac{9 \pi-\pi}{3}-2 \alpha\right)=\cos \left(3 \pi-\left(\frac{\pi}{3}+2 \alpha\right)\right)= \\
& =-\cos \left(\frac{\pi}{3}+2 \alpha\right)=-\cos \frac{\pi}{3} \cos 2 \alpha+\sin \frac{\pi}{3} \sin 2 \alpha=-\frac{1}{2} \cos 2 \alpha+\frac{\sqrt{3}}{2} \sin 2 \alpha
\end{aligned}
$$

$$
\cos \left(\frac{2}{3} \pi+2 \alpha\right)=\cos \frac{2}{3} \pi \cos 2 \alpha-\sin \frac{2}{3} \pi \sin 2 \alpha=-\frac{1}{2} \cos 2 \alpha-\frac{\sqrt{3}}{2} \sin 2 \alpha
$$

$$
X=\cos 2 \alpha-\frac{1}{2} \cos 2 \alpha+\frac{\sqrt{3}}{2} \sin 2 \alpha-\frac{1}{2} \cos 2 \alpha-\frac{\sqrt{3}}{2} \sin 2 \alpha=0
$$

Answer: 0.