# Change in angle with varying height (trigonometry)

I am working on a microscopy project and keep running into the same roadblock.

Some context: The microscope has a detector base with a variable distance from the source beam (you can move it up and down). There are 56 [0-55] distinct positions that the base can take. I know the angle of the beam across the base for the top and bottom positions but need to find a way to figure it out for all positions (this is dark field electron microscopy if you want to look into it more).

The question: Let me start with a picture

How can I determine what $\theta_3$ and $\theta_4$ are?

My trigonometry is not very strong so I am not sure if there is a relationship between the two which I am missing.

Progress so far: In addition to googling around, I have tried writing down equations for the situation but haven't gotten something I could work with yet. Here is a list of them.

$$tan(100) = (x+y)/a$$ $$tan(50) = x/a$$ $$tan(\theta_3) = (x+y)/(a+b)$$ $$tan(\theta_4) = x/(a+b)$$ $$tan(700) = (x+y)/(a+b+c)$$ $$tan(350) = x/(a+b+c)$$

As you can see this provides me with 7 variables and 6 equations, not enough to solve for anything.

I also looked into seeing if there was a relationship between the angle and step position (e.g. if linear: if you went to the middle, step 27, $\theta_4$ = 200rad and $\theta_3$ = 400rad) but I wasn't able to find anything online or on paper to convince me this would work.

• from the first two equations, you can have$y$ and so on. – hamam_Abdallah Oct 18 '16 at 16:41
• Your image seems to imply that you don't know how far you're moving the base at each step, and that the amount being moved is not consistent from step to step. If that's the case, then you just don't have enough information. If it's not, than some of your variables are actually known values. – Gabriel Burns Oct 18 '16 at 16:56
• @GabrielBurns I don't know how far each step is, but it's safe to assume it's the same distance each time. Using that I was able to figure it out. Thank you! – p.kubik Oct 18 '16 at 18:02