This would be absolutely awesome. I'm 14 and going into 9th grade. I'm taking algebra 2 / trig next year but I hate how they teach it so I use the internet to teach me. This would be a really cool idea, I'd definitely use it.
If you're interested in computer science, study topics in discrete mathematics, real algebra (the algebra where you compute with symbols instead of numbers, look into "rings", "groups", and "fields"), and some statistics. And if you have to scale back on something to make room, scale back on trigonometry.
Discrete is a subset of math that was barely touched on in my school, and it was a massive culture-shock to get to college and never have seen many of the staple discrete problems. Statistics is useful in real-time problem solving and analysis and is at the core of some of the neatest modern algorithms, including those in cryptography. Trig, on the other hand, is something that you will likely have pre-written libraries to support and is just not as important to most computer work. It's useful, but unless you're going into computer graphics and games it's maybe not as necessary.
These days, I can tell you the area of a triangle from its vertices but I still get thrown easily in a crypto discussion. If I had high school to do over, that's what I'd do differently.
From as practical as hammering wood together to as abstract as noticing the sine curves of the seasons.
And without Pythagorous, you can't grok Euler's Identity, which would be a real shame.
Finally, trig is the gateway to Euclidian Geometry, which is the first taste of realish math most people get in High School...
You can drop some of the petty stuff (x,y,(180-x-y) triangles), but don't skip the basics.
I'd also suggest skipping rings/fields since most of the algebra important to CS involves structures weaker than groups (semigroups and monoids). Also add graphs/combinatorics to your list.
[1] This fundamental fact is briefly mentioned in a subsection called "differentials", and otherwise ignored by textbooks.
When people argue about the math curriculum, they almost always argue the wrong question. The question is not, "Should we cover X?", because in isolation the answer is always yes! Should we cover trig? Yes! Should we cover set theory? Yes! Should we cover graph theory? Yes! etc. etc. The question is, "Given our limited time to allocate to math education, what are the best topics to focus on?", and once you consider the wealth of incredibly valuable topics neglected (elementary economics, elementary discrete math, actual algebra, game theory, computer programming, anything remotely resembling actual mathematical practices rather than memorized formulas stripped of all motivation and history), you'll find that spending umpteen weeks on trig is really shortchanging the students. The opportunity cost of trig is too high.
Let it be learned when it's actually useful. If you use it in physics, fine, learn it there, when you have context. Not in some abstracted "trig" course.
To be honest, I'm not sure I believe you anyhow. I took a lot of physics, as much as anyone not majoring in it will take, and I did not make heavy use of trig identities, nor did anybody else, nor do I recall a huge number of problems where they would have come in useful, and what problems they might have been useful in were textbook problems anyhow. (In the real world, inclined planes are not all at 30 and 45 degrees.) I think you might just be saying that to score rhetorical points.
If I had to learn trig my freshmen and sophomore year of college, when I was taking my introductory physics classes, I never would have kept up with the physics. My course assumed a solid foundation in trig and calculus.
I distinctly remember having to use various properties of triangles to solve many of my introductory mechanics problems.
You are probably right: when I teach calculus, there does seem to be an assumption that students know way too much petty nonsense. But some basics are necessary, even if the computer knows how to compute sin and cos. Students must understand angles and straight lines.
As for tradeoffs, I completely agree. I just think the value of basic trig is ridiculously high.
However, I worked as a math tutor for 4 years in college. I tutored both college students and in an after school program for east african immigrants.
What I observed in that experience was that people with good trig backgrounds did very well with the mechanical formula manipulation stuff in Calc. They were mostly having issues with proof structure, etc.
The students who had a sketchy trig background had problems with proof structure and also had a lot of issues simply doing mechanical symbol manipulation problems.
I majored in math and focused on abstract algebra. I don't think rings, groups and fields are very useful to general programming or even computer science. I actually have the exact opposite opinion on the matter. Unless you're going into crypto algorithm research you don't need to know that stuff, you just use a pre-existing library. On the other hand, trig establishes a foundation for stuff like robotics, DSP, computer graphics, computational physics and so forth. Almost anything falling into the traditional "applied math" bin requires you to have gotten trig down cold at some point.
However, arguing this is sort of a moot point because I can't think of any elementary or secondary school math sequence where you just skip over trig and take group theory instead. Trig is usually required somewhere along the way and abstract algebra is usually not available in high school unless you're going to a specialized math and science school. Indeed, it's usually not even available in college unless you're a math major.
If you have a sketchy trig background and need more practice and inspiration, I recommend:
Trigonometric Delights by Eli Maor, for inspiration.
Trigonometry Refresher by A. Albert Klaf for practice problems
Advanced Trigonometry by C.V. Durell and A.Robson for examples of advanced applications
Statistics is very important and totally overlooked by most hackers and even math majors (including myself.) If you have the opportunity to take a good stats class in high school or university, do so. I do not have any good recommendations on self study, but it seems like books on things like Biostatistics often lay the basics out more clearly than general stats textbooks.
Now.
learn to teach yourself math. I'd recommend getting a copy of mathematica and then find some good books on math subjects you're interested in and just get your learn on.
if you're getting into advanced algebra I'd recommend a bit of review of number theory and abstract algebra first, as that will make it much more interesting.