Derivation of Height Function given an angle

[Legaldose]

Hey everyone, I'm currently doing research at a University, I've been working on a problem for a few hours and wrote up a quick paper that shows my derivation of a certain height based on an angle. Basically the experiment is looking at optical properties of graphene, but for this to happen we need to shine a laser onto the species, where it reflects back into an analyzer. To get a measurement from a tiny change in angle, we will use a step motor attached to a threaded rod, and at the top of the rod somewhere is a winged nut that holds a wire attached on both sides, those wires run down to the laser and analyzer on each side. I wanted to find an equation that accurately measures the height of the winged nut above the machine as a function of the desired angle of incidence. The angle can only vary between 90 and 30 degrees. I will attach a couple of pictures to the machine, as well as my derivation, which includes a crudely drawn picture of the proposed step motor system. If anyone could go over my derivation it would be much appreciated. I already sent it to the PI to be looked over, but I just want to make sure this is correct, and what better way to do that than give it to you guys :p.

Also, I don't know exactly where this thread should go, so feel free to move it where it needs to go.

Double also, sorry for the thread title, it's 2:30am.

 

[LioNiNoiL]

From the diagram, the pertinent measurements are L and R₂ (which are fixed) and y which varies as the step motor is turned. The angle of interest (call it θ) is the angle opposite side L of the triangle formed by lengths L, R₂, and y.
By the Law of Sines, the ratio
L/sinθ = D (the diameter of the circumcircle about triangle yLR₂)
D = yLR₂/2A where A = area of triangle yLR₂ = √(s(s-y)(s-L)(s-R₂))
and semiperimeter s = (y+L+R₂)/2
from all of which we can get
θ = arcsin(2/yR₂)A