Why is $\tan 3 + \pi$ a near-integer?

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When playing with my calculator I found that $$\tan 3 + \pi \approx 3$$ Is there a mathematical reason for this?

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Expand $\tan (x)$ near $\pi$ I find $$\tan(x)=(x-\pi)+\frac{1}{3}(x-\pi)^3+O((x-\pi)^5)$$, and $abs(\frac{1}{3}(x-\pi)^3)\left.\right|_{x=3}<0.001$. I hope this might help