No solution for an arbitrary constant.

Say I have found a general solution to an equation of motion, which is, simple harmonic motion.

The general solution I have found is of the form:

$\displaystyle x(t) = L+B\cdot cos(\omega t)+C\cdot sin(\omega t)$

When inserting some (supplied) inital conditions, I am able to solve to find $\displaystyle B$ but cannot find a solution for $\displaystyle C$, does it follow that $\displaystyle C=0$?

My initial conditions contain only the position when $\displaystyle t=0$, $\displaystyle x$ and no derivatives of it - this being the case

$\displaystyle C\cdot sin(\omega t)=C\cdot sin(\omega \cdot 0)=0$

I hope it does, because I'm stumped if not - If the question above is too vague, I can supply the full question. However, I want to solve it myself and this is why I have used generic formulae above.