There is a natural isomorphism between the complex plane and the set of sine waves, but this isomorphism is ambiguous up to a rotation and/or flip of the plane. This ambiguity seems to be related to some of the differences in conventions about the Fourier transform and inverse Fourier transform.
In electrical applications, it's conventional that capacitive impedance is negative imaginary, and this requires that the cosine be represented by a point that lies counterclockwise from the one representing the sine. With this handedness, differentiation means multiplication by $i\omega$.
The convention I've been teaching is that $\sin\rightarrow1$ and $\cos\rightarrow i$, which is consistent with this. Call this convention A.
Looking at the form of Euler's equation, it does seem like it would be nicer to have $\cos\rightarrow1$. Sticking to the same handedness, we would then have to have $\sin\rightarrow -i$. Call this convention B.
Is this standardized? Is there a physicist's convention that is different from an electrical engineer's convention?
The WP article on the Fourier transform has some material that seems relevant, at "The reason for the negative sign convention in the exponent is that in electrical engineering, it is common..." This seems to imply that electrical engineers use convention B, but that other people use some other convention. Are the other people physicists? Some other kind of engineers? Do they use convention A?
Fourier transform standard practice for physics (I think what I'm calling convention A is not describable in the $(a,b)$ parametrization defined in jgerber's answer.)