Agree with the draw a triangle advice from Thorne. This is actually not so uncommon in books, though--just looked at a few. Maybe, a good pedagogy is to push it with students though, so they don't have as much memorizing rules.
Looked through a few books and most of the problems were just straight symbols are rather artificial word problems (volumes of revolution for given shapes).
One real area that they are useful is in hydraulics, especially for open channels (not filled pipes). Granville book has a short chapter on various hydraulics problems. Also Paul's online notes give an example.
Paul's: http://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx (towards bottom)
Granville: Sorry,no link. The free pdf doesn't have the same chapter...grr. My copy is 1944 edition.
There are several electrostatics problems requiring trig substitution. (Google search will show this, but I have a copy of Wangness in hand.) And electrostatics as taught in physics classes, not design work using FE, does a lot of analytical solutions in problems. Physicists like students to be able to work out some of the problems too--not just hand it to Mathematica. I would suspect that various applied problems in other engineering courses also occasionally have an integral needing trig substitution.
All that said, my advice would not be to teach this by physical examples. It will probably just make it harder. Treat it as another trick in the bag of tricks. If there are questions about application, you can cite some of these (read up on them a little bit so you are not just parroting something I say, but know it yourself Or just tell the kids, that there are a lot of integrals coming at them in physics and engineering courses and they need to have a decent toolkit.
[I think exposure to some decent variety of tougher integration methods, partial fractions and all that, makes students more confident in using integral tables also...at least they know they have seen a lot of stuff and worked with a lot of stuff. Not just blindly consulting the CRC. But this would be for kids who are taking a solid STEM course. Not a reduced calc class.]
Honest, I think part of the benefit of learning some things in math class is getting some of the learning in a more abstract fashion (with x's and y's) instead of physical variables. Let's the student learn it in that context. Then when they see it in physical science course or engineering, they learn it that way also.
Can be good to see things a couple different ways. Even if it is a year later, when they randomly see a trig substituation in a derivation or homework problem of a sci/eng class and they have forgotten it, they will remember they learned it once. So the discussion in sci/eng class will be easier than if this was first time. And the subsequent exposure may help solidify the concept or at least briefly refresh it and leave the student confident of knowing he could look up the method in his text or in a table of integrals.