Precalculus in reverse? In real analysis, I am aware of James Propp's "Real Analysis in Reverse" which does "naive reverse mathematics", showing the equivalence of various theorems of analysis to the completeness property of the real line.
Recently I've been thinking about the various geometric results that are typically covered in a precalculus course, such as:

*

*The Pythagorean theorem

*The law of cosines

*The equivalence between algebraic and geometric dot product ($u\cdot v = \|u\|\|v\|\cos \theta$)

*The formula for projections: $\operatorname{proj}_u(v) = \frac{v\cdot u}{\|u\|^2}u$

*The sine and cosine angle addition formulas

*The geometric interpretation of complex multiplication ("add the angles and multiply the lengths")

*Determinant of a $2\times 2$ matrix as the signed area of the parallelogram spanned by its columns

...and so on. It seems to me that these results are all saying "basically the same thing" (whatever that means); in particular it's often possible to prove one of them using some of the others.
I am wondering if there is a textbook or paper that does something like "naive reverse mathematics of precalculus". What would the "base axioms" be in this setting? What is the relationship between these theorems? Why are they so closely related? Are there other "equivalent" theorems that are missing from this list? Are there any keywords I should be using in my search?

From what I have seen here
(and I have contributed to this),
just about about any inequality
is a consequence of
that most elementary inequality,
Bernoulli's inequality.
