There is no uniform way of doing these things so I'm going to have to guess  do you mean why is there no cos(kx) involved, where k is whatever the constants work out to be?
Presumably the oscillating string has its end points fixed, so imgane extending the string into negative x  the resulting shape is an odd function  it has rotation symmetry of 180 degrees about the origin, so in its fourier expansion there will ony be odd terms and therefore no cos(kx)s involved.
