What will I learn in a trigonometry degree or training program?
The strict meaning of trigonometry is the study of triangles which doesn't sound all that interesting. However, nowadays we use it to mean geometry which is a much wider subject. You'll start out by learning the basic theorems in a degree or training program and then move on to their real world applications. These can be surprising, for example, all the excitement about nanotubes and buckyballs is based on their shape which is, you guessed it, a function of their geometry: indeed, it is people trained in geometry who were first able to explain why they are so special. The whole professions of cartographer (map maker) and surveyor are based on these trigonometric and geometric techniques that you would learn in such a degree or training program.
What would a trigonometry degree or training program qualify me for?
As with all other branches of mathematics a degree or training program in trigonometry can lead to any of a number of careers: from art (it's very important for understanding perspective) and architecture and astronomy (just to stay with the “a” s). The most likely careers are cartographer ($46,000 a year on average) and surveyor ($43,000). You can in fact say that these are the opposite ends of the same job: those who measure the world properly and those who prepare those measurements into the maps that we use.