OK. I've got a doctorate in math, and do research in geometry, and have looked at this stuff, and I've figured out what this guy is trying to accomplish.
It's a good idea.
It is NOT a good idea to teach INSTEAD of the standard geometry trigonometry, as long as all the rest of mathematics uses the standard functions. However, it IS a good way to teach the subjects of geometry and trigonometry themselves, as long as, at some point inthe course, the STANDARD definitons are introduced and their basic properties are proved.
Basically, the "quadrance" of a line segment is the square of its standard length, and the "spread" of an angle is the square of the sine of the angle. If you make these your fundamental quantities, most formulas and computations become simpler and their logical foundations become clearer. As long as, by the end of the course, students can translate freely between the two ways of looking at things, no harm is done, and a lot of conceptual and computational obstacles are avoided.
Almost every problem in standard trigonometry which requires a numerical solution must be solved by calculators or tables with messy approximations, even though the ultimate answer can be expressed in terms of the initial data by rational operations and square roots (and square roots are much, much, easier to calculate "by hand" than sines, cosines, and tangents). So Wildberger is onto something, though he's up against millennia of pedagogic tradition.
Doesn't this stuff sound an awful lot like vector cross products? Doing matrix maths on vectors can certainly be very useful (Quake does such maths in spades) but teaching it so early on seems premature.
BTW, if I were solving his example problem, I wouldn't use trig functions at all. I'd compute the area of the triangle, compute the altitude from that, and then have the slope of the upper-left edge. From that, I'd compute the intercept. The required length would follow naturally from that. No need for sines or other such functions (except that tan(45°)=1).
OK but how do you get the sine of the angle to square without trig - just wondering.
One use of trig is to transform equations into polar coordinate systems, where solutions are easier, depending on the geometry.
Can this be done using his approach?
IIRC, trig identities also were fundamental to much of calculus.
Using the basic trig formula sin(alpha+beta)= sin(45)=1/sqrt(2)
an exact answer can be obtained in about 6 short lines
without solving any messy equations
or using any of his pyrotechniques.