How Do You Derive Coefficients for Expanding f(x) Using Sine Waves?

[ChaseRLewis]

Teaching myself PDE so I'm understanding Separation of variables it's pretty simple up to the point you need to realize the orthogonality of the sin functions. So based on orthogonality it states that each constant becomes

A_n = 2 ∫₀¹ f(x)sin(nπx)dx

that f(x) isn't described very well and they just say "solved" so my guess is that f(x) is the initial condition such that

f(0) =initial at 0 x
f(1) = initial at 1 x
and nothing in between.

 

[Simon Bridge]

f(x) is an arbitrary function ... it looks like the relation is finding the coefficents for expanding f(x) as a linear combination of sine waves.