Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications.
Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?
Seriously, as a video game programmer, you use trig _every day_. If you want to find the distance between two points when you have the angle (eg. I'm firing a projectile at a certain angle), use cosine for the length of the x axis and use sine for the y axis. You get the hypotenuse from using pythagoras' theorem. If you want to find the angle between two points, use arc tangents.
Once kids have mastered all of that, you can teach them law of cosines and get into linear algebra; simple rotational matrices which can let you do collision detection and a whole bunch of other really, really cool stuff.
So yeah, Bob the builder might be boring as shit as a word problem, but Bob the builder might be a pretty cool video game. Without trig, as a game, Bob the builder is probably going to be boring as shit too.
Using cosine and sine and then Pythagorean theorem to get the length of the hypotenuse doesn't make sense - in order to get the length of the X and Y components with cosine and sine you need to already know the length of the hypotenuse.
I'm firing a projectile at 52 degrees and it needs to move 10 units. I'm at (5,7), where does the projectile end up?
delta x = cos(theta) * 10 delta y = sin(theta) * 10
So my final position is (5 + 6.16, 7 + 7.88), or (11, 15).
If I already knew how far the delta X and delta Y were, I could get the length of the hypotenuse with sqrt(6.16^2 + 7.88^2). That equals roughly 10, of course. If you want to try this out yourself, just remember that most trig functions are done in radians, so convert theta to radians with pi * theta (degrees) / 180.
It does, but for average Joe and Carla, who's day job involves making sure the box of sprockets sitting on the loading dock matches the P.O., and who's most pressing math problem is balancing their check book every few days and using turbo tax once a year, they'll never encounter anything more complicated than very basic arithmetic.
I think the hardest math problem most people encounter is figuring out percentages so they can fill out the "tip" on the credit card receipt when they eat out. Yeah it's "algebra" but it's a very limited class of problem and people just use a calculator or a tip calculator to do it these days.
>Trig
Yeah, trig really requires good motivation. The answer to most trig questions really is "buy the appropriate measuring tool" not "do a bunch of calculations and look up the result of the trig functions in a table". I don't ever remember having the calculations for cos(), sin() explained to me, they're just magic functions, number goes in, number comes out.
Unfortunately there's no easy, low-friction way for kids today to do that with their PC.
Being able to animate and play with your math helps a lot. GeoGebra is a good example, but it's not the holy grail.
You can also throw in some economics: Bob would be fired and/or get better equipment.