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Generally pre-calculus is taken after trigonometry, unless the trigonometry course was supplemented by a pre-calculus course, in which case the next course would be calculus.
That you aren't stuck with pre-cal.
At my high school, pre-cal mainly consists of trigonometry and advanced algebra and geometry. As the name suggests, it is very important for the preparation of entering a calculus course. In Calculus, using some, but not all, knowledge learned in pre-cal, you start to do things like solving derivatives and anti-derivatives. These help to solve instantaneous rate of change (or slope) of a curve, and the area under the curve, respectively, and much more advanced calculations. I think it is quite fun, though pretty hard sometimes. But then again I am a nerd.
Definitely AP Algebra (1)^2.
Pre-algebra, algebra, college algebra, geometry, and trigonometry