Finding the angle between 2 points on a circle

forgive me if this isn't the right place to ask this question but I am trying to figure out the value of theta along a line tangent to a circle from a starting position on the circle to an ending one outside of the circle. I have constructed an image to explain what I am trying to do. I know the value of the circles radius, the arc-length between the start and end point, and the height of the end point. With this information is it possible to determine the angle in question?

Thanks for the help! (Also not sure what to tag this question with)

Extend the green line to the centre of the circle $C$ and draw the radius of from $C$ to $P_{start}$
You now have a triangle $CP_{start}P_{target}$ where you know the length of $CP_{start}$ is $R$, the length of $CP_{target}$ is $R+a$ and the angle between them is $\frac{r}{R}$. So you can work out the other angles including $\theta + \frac{\pi}{2}$, for example by using the cosine rule to find the length of $P_{start}P_{target}$ and then the sine rule to find the angle. Then subtract $\frac{\pi}{2}$ to get $\theta$.