As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class....

I have to ask, do you realize how much of an outlier you are and how irrelevant your experience is to teaching normal kids mathematics? I mean... you immediately saw value in mathematically defining the relationship between the angle and the length. Teachers don't worry about kids like you (and me, and most of HN probably) because we'll be fine no matter how the class is taught. Wondering about how to get kids like you or me interested in math is like wondering how to get a cat interested in mice, or a Jersey Shore cast member interested in taking his shirt off. Teachers don't waste a second's thought on kids who will flourish regardless of the classroom environment. They worry about the marginal kids who might succeed if taught well but will flounder if taught poorly. A kid who really says "so fucking what?" to trigonometry is going to say "so fucking what?" to your trebuchet example, too. In the unlikely event he gives a damn about the length of the elastic rubber, he can just crank the attachment up and down and watch how the length changes, so who needs math?

The "best students" have sat through enough math classes that they know the right default assumption is that whatever is taught is probably useful. They get that engineering and science (including social science) depend on math, and math builds on previous math, and the high school curriculum doesn't have room for stuff that never comes up again later (okay, with the exception of geometry class, but even that is mostly useful.) Heck, even the "pretty good" students understand that. Those are the kids who get excited about real-world applications of trig: the kids who already get it and never ask "so fucking what?" Plus there's another group of kids who will never seriously ask "so fucking what," the ones who care about grades and college applications. They already know how they're going to use math: they're going to use it to get good grades in math class.

When students in trigonometry class really do say "so fucking what?" and "When are we ever going to use this?" they're always right. Because the kids who say that are the kids who really aren't going to use trig ever again in their lives, except maybe to scrape out a single math credit in college by taking a "College Algebra" course that's easier than the trig course they're taking in high school. The kids who say "so fucking what?" are the kids who are going to be business people, car mechanics, policemen, English teachers, coaches, corporate trainers, office workers, journalists, politicians, plumbers, soldiers, and grocery store managers. You know, the 98% of the planet that doesn't use trigonometry. There really isn't an answer you can give them. You can convince them that trig is essential for making video games, curing cancer, and any number of things they care deeply about, but what they're really thinking is, "I'm not going to be making video games. I'm going to be designing the ad campaign for the video game," or, "I'm not going to be curing cancer; I'm going to be mopping the floors at the hospital." Or, "I'm going to work for my dad for a few years at the dealership and then run for city council, and I'm going to pass a $5 million bond to hire an urban planning agency that probably employs some people who know trig."

I don't know how you motivate those kids to learn trigonometry, but you can't do it by convincing them they will apply trig in the real world, unless you are a very gifted liar ;-)