# 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?

• Calculus without trigonometry is a very weak system: almost nothing in life is a polynomial, which is all that can be integrated without knowing trigonometric and exponential functions; even something as innocent-looking as $\frac{1}{1+x^2}$ has a trigonometric-form integral. Commented Jun 16, 2016 at 14:15
• @Chappers The exact same criticism can be directed at basic algebra. My question is about the progression of teaching, not if calculus students shouldn't eventually learn trigonometry. Right now it looks like this super advanced topic that fear learning and many other sorts of classes (statistics, chemistry, etc) go out of their way to teach while avoiding any mention of calculus (even though all such students will have learned trig). Commented Jun 16, 2016 at 14:20
• Students need exposure to a broad class of functions first. This would include the trigonometric functions. Studying these functions in geometric terms first, rather than from a purely analytic viewpoint, is much more intuitive. Commented Jun 16, 2016 at 14:51
• Without trigonometry you don't have much interesting functions in calculus. Trigonometry provides a host of functions which serve as nice examples for applications of calculus. BTW basic calculus is not easier than basic trigonometry because basic trigonometric essentially deals with algebraic properties of circular functions. Calculus on the other hand is essentially non-algebraic in nature. Commented Jun 16, 2016 at 16:38
• Maybe a better place for this question would be matheducators.stackexchange.com Commented Jun 17, 2016 at 21:05

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.

• "You can't just deal with this integral as if it's ∫f2(x) dx∫f2(x) dx where ∫f(x) dx∫f(x) dx is known because that's not how integrals work. We need to use our trig identities to make this easier:" Yes, but I do the same in algebra. I don't treat them like generic functions and I depend on their identities in both contexts. Commented Jun 16, 2016 at 21:34
• @CircleSquared This means I misunderstood the statement I quoted you on at the top. What did you mean by that? Commented Jun 16, 2016 at 21:35
• "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." Why? We do exactly that with pre-trig algebra. Commented Jun 16, 2016 at 21:35
• @CircleSquared Pre-trig algebra moves at a really slow pace because students aren't experienced. I'm probably biased here because I'm a high school student, so all of the calculus students I know are faster learners. In any case, given that Calc I with trig functions is a whole course and that trig is a pretty big part of Calc I, I don't think Calc I without trig can become a whole course. I'm not a teacher, so I might be wrong about whether or not there could still be enough for a full course without trig, but I know that leaving trig out of calc would leave much curriculum out of the course. Commented Jun 16, 2016 at 21:39
• It should be mentioned that, frequently, the first nontrivial limit one encounters is $\lim_{x\to0}\frac{\sin x}{x}$, so it's good to have available at least some basic trigonometry. Circular functions provide other several interesting examples, such as $f(x)=x\sin(1/x)$ and friends. Commented Jun 17, 2016 at 21:04

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.