Period of sinusoid function, smallest value when it repeats itself

Question

Okay, so suppose you have f(t) = sin($400 \pi$ t)

Period will be, $T = \frac{2 \pi}{w}$ = $\frac{2 \pi}{400 \pi} = \frac{1}{200} s $

So for t = 1

whatever value of sin($400 \pi \cdot 1$) is should be the same as sin$(400 \pi \cdot 1 + \frac{1}{200}) $

shouldn't it? Except it's not!

sin($400 \pi \cdot 1) = 0$

sin$(400 \pi \cdot 1 + \frac{1}{200}) = 0.00499$

?????

Answer 1

Note that :$$\color{red} {f(t+T)=f(t)} \\T=\frac{1}{200} \\\to\\f(t+\frac{1}{200})=\sin(400\pi(t+\frac{1}{200}))=\\\sin(400\pi t+2\pi)=\\\sin (400\pi t)=\\f(t) \color{red} {\checkmark}$$