I know it's a side point, but I loved the comment: "...explaining for instance why in any trigonometric identity such as sin(x+y) = sin(x) cos(y) + cos(x)

I know it's a side point, but I loved the comment: "...explaining for instance why in any trigonometric identity such as sin(x+y) = sin(x) cos(y) + cos(x) sin(y) the number of odd functions (sine, tangent, cotangent, and their inverses) in each term has the same parity." I never thought of it that way. I always converted to exponential notation to derive them, but you could use this units approach to provide what you needed.