Finding $\theta$ in this geometric construction

Question

I am working on a laser wavefront analyser and I need to calculate the angle of tilt of a diffuser in my setup which I illustrated below :

The screen is shown in green an the angle I want to calculate is $\theta$, in blue. I know $\alpha$, $R_1$, $R_2$, $d_1$, $d_2$ and $x$, but I don't know $R_x$.

From basic geometric relations, I was able to pull these three relations, but I seem to miss at least one to complete the puzzle.

$$\tan(\theta)=\frac{R_x}{d_2+x}$$ $$\frac{d_2+x}{R_1+R_2+R_x}=\tan(\alpha)$$ $$\tan(\alpha)=\frac{x}{R_x}$$

So, what is $\theta$?

Answer 1

Inverting your second equation gives

cot(α) = Rx / (d2 + x) + (R1 + R2) / (d2 + x)

thus

Rx / (d2 + x) = cot(α) - (R1 + R2) / (d2 + x)

and from the first equation this is tan(θ) in terms of α, R1, R2, d2, x.