Math and Metal
My current day job is a sheet metal tech. Every day, I slice slabs of metal into thinner slabs and bend them to specific instructions. The average person might call it necessary blue-collar drudgery. I see opportunity to re-explain different aspects of knowledge.
For instance, the average slab of metal comes in a 10-foot by 4-foot sheet. Commonly, we're tasked to slice along the length to either 4 inches, 6 inches, or 9 inches of length. The excess metal is scrapped.
Now, /ideally/, if I'm given an order for 6-inch pieces the 4 feet (48 inches) should be sliced into 8 equal pieces. Unfortunately, our slicer is not ideal; it cuts at a slight angle such that when it is set to 6 inches, the final piece is almost always an eighth of an inch short on one end and long on the longer. Knowing that I've made 7 cuts, I can calculate the angle the blade actually cuts.
We know that the sheet is ten feet long, and has a difference of half an inch in height after 7 cuts. I can rewrite that as a triangle with length 120 and height .5. If you're decimal or fraction-averse, it's the same as having a triangle with length 240 and height of 1. The ratio between the distances remains the same, so the angles won't change. (Side note: angles can also be described as ratios between length and height, though it's uncommon.) Since we only have the length and height of a triangle to go on, the tangent is the only tool immediately available. Sure, you could calculate the hypotenuse to use sine/cosine, but it's extra unnecessary work. Plus, you'd probably end up with enough rounding error to make it not worth your while.
So the tangent of the angle is 1/40. Thankfully, we have computers now that will do the number-crunching in order to find what that angle is. Plug it into Wolfram Alpha and I get an angle of about 1.4 degrees. Which, mind you, is the result of seven cuts. Divide by 7, and we find out that the actual blade is only off by 0.2 degrees. Such a small difference, but such a large effect of making the last sheet unusable!