$\begin{array}{l}\text { 3. } \cos \frac{\pi}{11}-\cos \frac{2 \pi}{11}+\cos \frac{3 \pi}{11}-\cos \frac{4 \pi}{11}+\cos \frac{5 \pi}{11} \\ = \\ \text { (answer with a number). }\end{array}$
3. $\frac{1}{2}$.

Notice that,
$$
\begin{array}{l}
\sum_{k=1}^{5}(-1)^{k+1} \cos \frac{k \pi}{11} \\
=\frac{1}{2 \cos \frac{\pi}{22}} \sum_{k=1}^{5}(-1)^{k+1}\left[\cos \frac{(2 k+1) \pi}{22}+\cos \frac{(2 k-1) \pi}{22}\right] \\
=\frac{1}{2 \cos \frac{\pi}{22}}\left(\cos \frac{\pi}{22}+\cos \frac{11 \pi}{22}\right)=\frac{1}{2} .
\end{array}
$$