I am trying to find the side lengths of a rectangular prism. The diagonal length is 1 and I know the angles connecting the diagonal corners of each side. Here is a diagram: 
My question is: remembering that the sides x, y, and z are not equal, what are their lengths with respect to angles α, β, and γ?
HINT: since the diagonal has length $1$, we have $$\sqrt {x^2+y^2+z^2}=1$$
Based on your figure, we also have $$x=\sqrt {x^2+y^2} \cdot \cos \alpha$$ $$z=\sqrt {y^2+z^2} \cdot \cos \beta $$ $$x=\sqrt {x^2+z^2} \cdot \cos \gamma$$ Now solve this system. Also note that three of these equations are sufficient to determine $x,y,z $. This means that once fixed two angles, the third is automatically determined.
$$x^2+y^2+z^2=1$$ $$x^2:y^2=ctg^2α$$ $$y^2:z^2=tg^2β$$ $$x^2:y^2:z^2=ctg^2 α:1:ctg^2β$$ $$x=\frac{ctgα}{\sqrt{ctg^2 α+1+ctg^2β}}$$ ...
