What a great read! But the staircase example seems overwrought.
Reality often doesn't need to be that precise. If you're a few degrees off in your angle, it won't matter. The floor and/or walls are probably not perfectly flat and may not even be perpendicular. Stairs will work anyway.
This angle problem has been solved in tooling. You set the (<$100) chop saw for 30° (or whatever), mark the length of the board, and cut. It takes seconds, no tracing required. Trig is a great solution, especially given the "close enough" tolerances of wood.
Notching angles into boards is the reason framing squares exist. I've learned and forgotten this three times in my life, the last not so long ago.
Stairs are for human legs, and they care about the rise, the run, and the depth of the stair (tread). And all those can be done with a framing square. Angles are incidental to this (google around and you'll see a lot of stringers being cut with a circular saw).
A 3/5 ratio is 31°. You slap down the square, you measure 3 inches on one arm and 5 on the other, and you mark it with a pencil.
Random dude answering question on the internet:
> In US construction stair pitch (angle) is expressed as a ratio between rise and run (vertical and horizontal distances).
> For commercial construction the minimum ratio is 7:11 but 6:12 is often used (these are expressed as rise and run in inches which is why it’s 6:12 and not 1:2)
Maximum ratio. In the ("whole") rest of the world it is probably 15:30 to 17:26.
Crappy tooling, and crappy software in general, is our legacy from Microsoft. Before Microsoft, people had expectations that software would work right. It didn't always, but it was a big deal when it didn't. Microsoft, single-handedly, got people used to rolling their eyes and rebooting, and not demanding a refund, partly by never ever refunding anything. It worked for them, and here we are.
The avoidance of detail, enabling its occasional rediscovery today, is equally new. People took pride in attention to detail, and demanded it. Watch any detective movie from the 30s or 40s and you get a glimpse of how attention to detail was respected.
Reality often doesn't need to be that precise. If you're a few degrees off in your angle, it won't matter. The floor and/or walls are probably not perfectly flat and may not even be perpendicular. Stairs will work anyway.
This angle problem has been solved in tooling. You set the (<$100) chop saw for 30° (or whatever), mark the length of the board, and cut. It takes seconds, no tracing required. Trig is a great solution, especially given the "close enough" tolerances of wood.
If you need precision, work with metal.