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Eigenfunctions of the Sine Function

Try it yourself: consider the function f(x) = sin(kx).

For what values of k is f periodic with period 2p? That is, For which values of k is f(x+2p) = f(x), or, when does sin(k(x+2p)) = sin(kx)? (Distribute the k and use the formula for the sine of a sum of angles on the left-hand side.)

What eigenvalue would this give? E.g. what m makes f(x) = m f''(x) for a given k?

Eigenfunctions of the Laplacian