3. Calculate the value of the expression $\sin \frac{b \pi}{36}$, where $b$ is the sum of all distinct numbers obtained from the number $a=987654321$ by cyclic permutations of its digits (in a cyclic permutation, all digits of the number, except the last one, are shifted one place to the right, and the last one is moved to the first place).
3. Solution. When summing all different numbers obtained from the number $a=987654321$, including the number $a$ itself, in each digit place we will get the sum of all its digits $9+8+7+\ldots+3+2+1=45$. Therefore, $b=45\left(10^{8}+10^{7}+10^{6}+\ldots+10^{2}+10+1\right)=45 \cdot 111111111=4999999995$. Calculate $\frac{\pi b}{36}=\frac{\pi \cdot 45 \cdot 111111111}{36}=\frac{\pi \cdot 555555555}{4}=138888888 \pi+\frac{3}{4} \pi$. From this, it follows that
$\sin \frac{\pi b}{36}=\sin \left(138888888 \pi+\frac{3}{4} \pi\right)=\sin \frac{3 \pi}{4}=\frac{\sqrt{2}}{2}$.