Ghee, a clarified butter commonly used in Indian cuisine, typically contains around 80% milk fat. The remaining portion includes water and milk solids that are removed during the clarification process.
In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $\angle OPQ = 15^\circ$ and $\angle PTQ = \theta$, then find the value of $\sin 2\theta$. 