I'm trying to wrap my head around the fact that multiplying anything on a sine wave then calculating the sum of products is a sine of the sine wave's phase.
This is literally the foundation of the Fourier transform, because it allows us to find the maximum of the signal's cross-correlation with a sine wave by sampling it (cross-correlation) in just two points separated by 90 degrees (as opposed to analyzing the whole cross-correlation curve as I do in the following plots).
Intuitively I do understand that this has something to do with the fact that cross-correlation has integration under the hood, and a sine stays a sine after being integrated. But a more illustrative explanation would be nice.
I searched the Web and the only explanations that pop up are based on Fourier transform. That won't do.
Poor man's equation:
$(f(t)\star\sin(\omega t))(\tau)=A\sin(\tau+\phi)$
Call $$y(\tau) = (f(t) \star \sin (\omega t))(\tau)$$ Then, using differentiation under integral sign you can check that $y$ satisfies the differential equation $$y'' = - \omega^2 y$$ whose solution is well known to be a sine function.