Analytical IK for simple 2 DOF robot

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Aware this isn't a ROS question; but thought there may be good overlap with the know-how of this community. Happy to be advised if this is not a good use of this board.

I'm trying to come up with an analytical solution for inverse kinematics for my robot; but some of my trig has left my brain!

The robot is very simple; with one rotary and one prismatic joint. I'm looking to calculate the rotation; which is t.

I've got the below equation which gives me the z co-ord from t and e & w which are constants that I need to tune the robot.

z = e*sin(t) + cos(t)/sin(t)*(-x - e*cos(t)) + w

So basically; I just need to solve the above for t.

I did the below attempt at simplification; and then I get stuck.

z = e*sin(t) + cos(t)/sin(t)*(-x - e*cos(t)) + w
z = e*sin(t) - x*cos(t)/sin(t) -e*cos^2(t)/sin(t) + w
z - w = (e*sin^2(t) - x*cos(t) - e*cos^2(t) ) /sin(t)
z - w = (e*(1 - cos^2(t)) - x*cos(t) - e*cos^2(t) ) /sin(t)
z - w = (e - e*cos^2(t)) - x*cos(t) - e*cos^2(t) ) /sin(t)
z - w = (e - 2e*cos^2(t)) - x*cos(t) ) /sin(t)

Is this going down the correct path to solve? How do I go from here toward getting to the form t = ...

Thanks