Graham, in five years of observing workers (skilled and unskilled) at
volunteer construction sites I have yet to see a single example of someone
using trig methods to solve problems involving angles. I watched, for
example, a highly-competent professional carpenter walk a novice through
the process of laying out a roof truss basically as a geometry
construction, using a rafter square and a pencil. The carpenters that I
have observed rarely do anything with angles, preferring to convert things
like roof slopes into linear rise/run measurements.
I don't think this means that math beyond algebra and geometry is a waste
of time, however. If nothing else, there is some value in knowing that
there is another way of solving problems, even if it requires things like
protractors and trig tables that might not be ready-to-hand. Further,
there is power in knowing about these other tools, even if you never have
cause to actually use them---if only to keep them from being the exclusive
domain of the engineers and architects on the job.
This is not to say that the way that we currently teach things like
trigometry and calculus is optimal. Having students do endless drill work
on de-contextualized problems and memorizing formulas only to forget them
immediately after testing seems pretty useless. Maybe if we engaged
students in things like designing roof trusses in school, they would have
an easier time seeing learning trig as an empowering skill. ---Tim