Why are sine and cosine vectors? I understand that sine and cosine functions are vectors, and I understand that what defines a vector is how it transforms under change of basis. Am I correct in understanding that means that for some angle $\alpha$ in the $(x,y)$ coordinate system and some $\alpha'$ in the rotated coordinate system $(x',y')$ $$\sin(\alpha')=\frac{\mathrm{d}\alpha}{\mathrm{d}\alpha'}\sin(\alpha)?$$

I've tried working with this definition for the transformation, which I've been led to believe is correct, but I can't seem to show that it holds. Is this even a correct starting point? If not, what is?
The word "vector" means slightly different things in various contexts. In linear algebra, a vector space is a structure satisfying a certain list of axioms, and a "vector" is just an element of whatever vector space you're talking about.

In that sense $\sin$  and $\cos$ are vectors; they're elements of for example $C(\Bbb R)$, the space of all continuous functions on the line. But this use of the word "vector" has more or less nothing to do with the thing you understand about how a "vector" is defined by how it changes under coordinate transformations.

That second sense of the word comes up in differential geometry, where for example it distinguishes vectors from "co-vectors".

I may be oversimplifying, but I believe that the word is being used in two different ways in "$\sin$ is a vector" and "what defines a vector is how it transforms under change of basis".
The word "vector" means slightly different things in various contexts. In linear algebra, a vector space is a structure satisfying a certain list of axioms, and a "vector" is just an element of whatever vector space you're talking about.

In that sense $\sin$  and $\cos$ are vectors; they're elements of for example $C(\Bbb R)$, the space of all continuous functions on the line. But this use of the word "vector" has more or less nothing to do with the thing you understand about how a "vector" is defined by how it changes under coordinate transformations.

That second sense of the word comes up in differential geometry, where for example it distinguishes vectors from "co-vectors".

I may be oversimplifying, but I believe that the word is being used in two different ways in "$\sin$ is a vector" and "what defines a vector is how it transforms under change of basis".