> Perhaps they'd say those are the simplest, purest instantiation of a frequency.
Actually I thought it is the other way round: A frequency is defined as a sine basically, it has an
* amplitude
* frequency
* phase shift
The phase shift is why we need cosines and sines in the Fourier transform as cosine is just a shifted sine.
The frequency is the constant in the argument of the sine. And here is where the tail might chase the dog: it's the definition, not an observation I'd think.
> When performing a Fourier transform, we represent the signal in a new basis, where each component
... each component is a sine, that is described by the 3 constants above.
> In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are Sines and cosine.
Exactly, a complex exponential function is just that spring. And if the absolute value of the argument is not exactly one, the spring spirals outwards or inwards.