Why all near integers

Question

$C$MRB is a symbol for the MRB constant.

Why do I get all near 0, 1 or 2 for all values of n for sin(Pi/$C$MRB*(5060936308 + 78389363/24*Floor(n)))? The results near 0 are very small and the arguments are all very large!

With Mathmatica let m be $C$MRB:

m = 0.187859642462067120248517934054273230055903094900138786172004684089477231564660213703296654433107496903

N[Table[1 + Sin[Pi/m*(5060936308 + 78389363/24*Floor[n])], {n, -100, 
    100}], 20] // TableForm

gives the following 4,800 near integers. Can you explain why? Finding near-integers in a range might sheed some light on this.

{
 {2.5774797752962409150*10^-15},
 {0.99999992891277049619},
 {1.9999999999999975236},
 {1.0000000696655945452},
 {2.3773806400995703132*10^-15},
 {0.99999993175604041349},
 {1.9999999999999977196},
 {1.0000000668223246279},
 {2.1853656887255180674*10^-15},
 {0.99999993459931033079},
 {1.9999999999999979076},
 {1.0000000639790547106},
 {2.0014349211740841794*10^-15}
}
...
{
 {2.0498172895263341762*10^-17},
 {0.99999999430797859407},
 {1.9999999999999999876},
 {1.0000000042703864473},
 {6.3352656188152076990*10^-18},
 {0.99999999715124851137},
 {1.9999999999999999977},
 {1.0000000014271165300},
 {2.5654216498544892548*10^-19},
 {0.99999999999451842867},
 {1.9999999999999999998},
 {0.99999999858384661268},
 {2.2620025337740654904*10^-18},
 {1.0000000028377883460},
 {1.9999999999999999937},
 {0.99999999574057669538},
 {1.2351646725181057378*10^-17}
}
...

Answer 1

$$\frac{5060936308}{C_{MBR}}\approx26939986905.5000000002\approx\frac{53879973811}{2}=a$$

$$\frac{78389363}{24C_{MBR}}\approx17386509.4999999997\approx\frac{34773019}{2}=b$$

So we basically have

$$ \sin\left(a\pi+b\pi n\right)=\sin\left(a\pi\right)\cos\left(b\pi n\right)+\cos\left(a\pi\right)\sin\left(b\pi n\right) $$ $$ =-\cos\left(b\pi n\right) $$ which is cyclic as you can see here.

NOTE:

$$ \sin\left(a\pi\right)=-1,\cos\left(a\pi\right)=0 $$