Does sin(integer) series repeat itself?

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sin(1) = 0.84, sin(2) = 0.9, sin(3) = 0.14............sin(314)=0.15...and so on..

I need to find an integer after which sin series starts repeating itself for subsequent integers. I need this as I want to create series as sin(x+RANDOM) where RANDOM is a random number and I would choose this random number from the range where this series starts repeating.

Does sin(integer) series repeat itself after any interval?

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As you are working in radians, there is no integer where $\sin(x)$ starts repeating for subsequent integers. However if you are able to work in degrees, then the sequence starts repeating right from the beginning, with period $360^{\circ}$. So $\sin(0^{\circ})=\sin(360^{\circ})$.