Number of solution for $\sin x = 0$ in $[0,2π]$ are 2 or 3

Question

Number of solution for a equation $\sin x = 0$ in $[0,2π] $ (close interval) are 2 or 3. Solutions are of course 0,π,2π but are they three solution or just two solution considering 0 and 2π are same thing for most part? So what is the right answer 2 or 3.According to me answer should be 3 but my book said 2.

Answer 1

$0=0.0000000\cdots$ and $2\pi=6.2831853\cdots$, I wouldn't consider that they are "the same thing", even "for most part".

There must be a misunderstanding about what your book says.

Answer 2

Indeed:

$$\sin(0) = 0 $$

$$\sin(\pi) = 0$$

$$\sin(2\pi) = 0$$

Thus there are $3$ solutions.

It doesn't really make sense to argue that $0$ and $2\pi$ are the 'same' because they are different numbers? Yes if we know that $0$ is a solution, then so is $0+2k\pi$ on a suitable domain - but the fact that we exploited periodicity here doesn't make our other solutions more or less different!