3. Find the rational number - the value of the expression

$$
2 \cos ^{6}(5 \pi / 16)+2 \sin ^{6}(11 \pi / 16)+3 \sqrt{2} / 8
$$
Solution: By the reduction formula, the desired expression is equal to

$2 \cos ^{6}(5 \pi / 16)+2 \sin ^{6}(5 \pi / 16)+3 \sqrt{2} / 8$. By the sum of cubes formula, we have:

$\cos ^{6}(5 \pi / 16)+\sin ^{6}(5 \pi / 16)=1-\frac{3 \sin ^{2}(10 \pi / 16)}{4}$. Therefore, our expression takes the form:

$2-\frac{3 \sin ^{2}(10 \pi / 16)}{2}+\frac{3 \sqrt{2}}{8}=2-\frac{3}{2}\left(\frac{1-\cos (5 \pi / 4)}{2}\right)+\frac{3 \sqrt{2}}{8}=\frac{5}{4}$. Answer: 5/4.