calculating approximate solutions to DEs

Question

I frequently come across equations of the form: $$dy=\sin(d\theta)+...$$ where $y$ is a function of $\theta$. My lecturers normally bypass this by making the assumption that $\sin(d\theta)=d\theta$ and $\cos(d\theta)=1$. Whilst I agree that this is close, exactly how close are these solutions? I am looking for examples of it being solved exactly to see how big the errors are.