Finding a basis of a subspace. Can you please help me with this task?

Let $f_1=\cos 2x,f_2=\sin x,f_3=\sin^2x$. Consider the linear space $\mathbb{W}=\left\langle f_1,f_2,f_3\right\rangle$.
Let $\mathbb{U}=\{f\in\mathbb{W}|f(0)=0\}$. Find a basis of this subspace.

I have proved that $f_1, f_2, f_3$ are linearly independent by considering the equation
$$a\cos(2x)+b\sin(x)+c\sin^2(x)=0$$
How do I find a basis? Should I substitute $0$ into $x$?

To summarize the discussion in the comments:
This problem is somewhat easier than the general problem of this type because it happens that a good basis for $\mathbb U$ is a subset of the given basis for $\mathbb W$.
Indeed, both $f_2$ and $f_3$ are in $\mathbb U$ so $\mathbb U$ contains the $2$ dimensional subspace spanned by $f_2,f_3$.  Now that has to be all of $\mathbb U$ since if $\mathbb U$ contained anything else it would have dimension $3$ hence would be all of $\mathbb W$, but $f_1\not \in \mathbb U$ so this is not possible.
In general, there is no reason for a basis of $\mathbb U$ to be a subset of the given basis, but it is the case here.
