How to estimate this special integral?

Question

Let $\theta\in(0,\pi)$, and $$ {\rm I}\left(\lambda,\theta\right) = \int_{\theta}^{2\pi - \theta} \left[({1 \over \sin\left(t\right)}\frac{\partial}{\partial t})^{k}{\rm e}^{\left({\rm i}\lambda - 1\right)t}\right] {\sin\left(t\right) \over \sqrt{\cos\left(\theta\right) - \cos\left(t\right)\,}\,} \,{\rm d}t $$ I want to know the asymptotic behavior when $\theta$ near $0$, and I also want to know if there is a more clear expression of this integral. I guess it may relate to the Bessel function. Thanks in advance.