Why is calculus normally taught after trigonometry (instead of more immediately after algebra)? This question is a little meta. I hope I'm in the right place.
In my experience the teaching of calculus is normally delayed until after learning basic trigonometry. Now that I've started learning calculus, this seems odd. It seems as though calculus applies trigonometric functions in the same way algebra does, i.e. it's a system of operations which doesn't depend on things like trig' in any way, but rather serves as a foundation/context for using trig'.
Actually, it seems like basic calculus is easier than basic trigonometry. Am I missing something? Or, is the conventional progression of teaching mathematical topics more of a history lesson?
 A: 
It seems as though calculus applies trigonometric functions in the same way algebra does, i.e. it's a system of operations which doesn't depend on things like trig' in any way, but rather serves as a foundation/context for using trig'.

I'm not exactly sure what you mean by this, but calculus does not rely on only basic trig. Double-angle, half-angle, sum to product, and product to sum are all important to understanding calculus and doing a lot of calculus integrals. The trig functions aren't just treated as generic functions with certain derivatives and integrals to be memorized. For example:
$$\int \cos^2 x \ dx$$
You can't just deal with this integral as if it's $\int f^2(x) \ dx$ where $\int f(x) \ dx$ is known because that's not how integrals work. We need to use our trig identities to make this easier:
$$\int \frac{1-\cos 2a}{2}dx$$
Now, this integral is a lot easier than trying to treat $\cos x$ as just a unit without regard for trig identities.
Finally, yes, for the derivatives and integrals of polynomials in basic calculus, you do not need trigonometry. However, I don't think people can make a whole course out of just limits and then limiting the scope of derivatives of integrals to outside trigonometric functions. On the other hand, we could teach basic derivatives and integrals first and then move onto trig, but this would either mean that people would be learning trig identities at the same time as they were learning trig calculus, which would probably overwhelm students, or an interruption in calculus curriculum in order to learn trig, which would probably cause students to forget basic calculus. Most people take pre-calculus then calculus because they know their trig identities from pre-calculus and then those trig identities are reinforced during calculus, so they are not forgotten while they go through their calculus course.
A: Personally, I remember learning about trig functions in Algebra II/Trig and rational functions (as well as polynomial division) in Precalculus. Tangent functions and rational functions have asymptotes, the understanding of which is important in the study of limits. In addition, having practiced with all kinds of functions, concepts like "instantaneous rate of change" and "area under a curve" were easier to digest. 
