Suppose I have a $d$-bit estimate of $\theta \in (0, 2 \pi]$, say a register of qubits $|\tilde{\theta} \rangle = |z_1 \rangle \ldots |z_d \rangle$ for $|z_i \rangle \in \{|0 \rangle, |1 \rangle \}$ where $\theta \approx \sum_{i=1}^{d} z_i \pi /2^{i}$, and I wish to implement a transformation $|\tilde{\theta} \rangle |0 \rangle \mapsto |\tilde{\theta} \rangle (cos(\theta) |0 \rangle + sin(\theta)|1 \rangle)$ using a quantum circuit consisting of two-qubit gates which can be represented by the controlled unitary transformation
$U_{c_i} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos(\pi / 2^i) & -sin(\pi / 2^i) \\ 0 & 0 & sin(\pi / 2^i) & cos(\pi / 2^i) \\ \end{bmatrix}$
I am trying to come up with a formula for how a sequence of these controlled operations would be applied to the ancilla qubit $|0 \rangle$ (maybe more than a single qubit?) to result in an overall rotation by $\approx \theta $ but I am not sure how to proceed. Any insights appreciated.
Edited the matrix as in the answer.