Hi Raj,
There is an error in the figure you have attached the reference of $\displaystyle \theta$ as represented by an arc is a mistake it complicates things. I have modified the figure I hope this is what you intended to ask.
In the figure attached the angle suptended at the circumference is $\displaystyle \theta$ (by alternate segment theorem). So we have $\displaystyle \theta$ = $\displaystyle sin^{-1}(\frac{L}{R})$
Kalyan.