This image represents the starting configuration, looking down at the $x$-$y$ plane.
Point $p$ starts at the specified coordinates in the $x$-$y$ plane.
The first 'axis' is located in the $x$-$y$ plane, rotated 35° about the $z$-axis. The point is rotated about this axis by angle $θ$ such that it moves 'up' in the $z$ direction.
The point is then rotated about the x-axis by another angle $α$, again moving 'up' in the z direction.
I am trying to create a function $z_2(θ)$ which gives the z coordinate of the point's final position in terms of $θ$, which is my independent variable. All other values are known constants. I don't know if it helps to know the actual values, but here they are:
$x_1 = 0.005216$
$y_1 = 0.005224$
$α = 7.902869°$
The value of $θ$ will be small, roughly between 5° and 15°. Neither rotation will move the point out of the first quadrant where it is initially depicted.
Is it possible to solve this with trigonometry? I am hoping to avoid needing quaternions.