Define $p_n = 5^n \sin(n\alpha)$ and $ q_n = 5^n \cos(n\alpha)$.
Then we have:
\[p_{n+1}=5^{n+1} \sin ((n+1)\alpha )=5^n\sin (n\alpha )5\cos (\alpha )+5\sin(\alpha )5^n \cos(n\alpha ) =\pm 4p_n+3q_n \\\\ q_{n+1}=5^{n+1}\cos ((n+1)\alpha )=5^n\cos (n\alpha )5\cos (\alpha )-5\sin(\alpha )5^n \sin(n\alpha ) =-3p_n\pm 4q_n\]
Now, $p_1 = 3$ and $q_1 = \pm 4$ are integers, and on the second level we get a new pair of integers:\[p_2 = \pm 4 p_1+3q_1 = \pm 4\cdot 3+3\cdot (\pm 4) = \pm 24 \\\\ q_2 = -3p_1 \pm4 q_1 = -3\cdot 3\pm 4 (\pm 4)=7\], so by induction the $j$th level also is expressed by integers, hence $a = p_{1000}$ is an integer.
Now, define $r_n = p_n \:\: (mod\:\:5)$ and $s_n = q_n \:\: (mod\:\:5)$.
($r_1 = p_1 \:\: (mod \:\: 5) = 3$ and $s_1 = q_1 \:\: (mod \:\: 5) = \pm 4 = \mp 1$).It follows, that
\[ r_{n+1} = \pm 4p_n+3q_n \:\: (mod \:\: 5) = \pm 4 r_n + 3s_n = \mp r_n+3s_n.\]
and
\[s_{n+1} = -3p_n \pm 4 q_n \:\: (mod \:\: 5) =-3r_n \mp s_n.\]The two modulo sequences (for $q_1 = \pm 4$) are periodic and both have period $4$, and we get:
$q_1 = 4$: $r_1 = 3, r_2 =4 , r_3 =2 , r_4 =1 , r_5 =3, r_6 = 4, …$
$q_1=-4$: $r_1 = 3, r_2 =1 , r_3 =2 , r_4 =4 , r_5 =3, r_6 = 1, …$
- hence
\[a\: \: \: (mod\: \: 5)=\left\{\begin{matrix} 1,\: \: q_1 = 4\\4,\: \: q_1 = -4 \end{matrix}\right.\]