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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 $\sin 2\theta$.
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$\angle QPT = 75^\circ$ ($\frac{1}{2}$ mark). $\angle PQT = 75^\circ$ ($\frac{1}{2}$ mark). $\theta = 30^\circ$ (1 mark). $\sin 2\theta = \sin 2(30^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}$ ($\frac{1}{2} + \frac{1}{2}$ marks).