I have just seen the proof for Euler's theorem, and i have to say, i am pretty amazed! But I still don't understand why it is useful, or intuitive, at all. Idk where to even begin. I have seen the wiki page, but it is either convoluted or seemingly useless.
Thanks
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$\begingroup$If we call Euler's formula $\exp(i\theta)=\cos(\theta) + i \sin(\theta)$ then one of the more esoteric reasons to use Euler's formula is to represent rotations easier. Rather than representing a rotation as a $2\times 2$ matrix multiplication with a two element vector, we can represent it as a multiplication of a complex exponential $\exp(i\theta)$ with a complex number. This can simplify equations involving rotations drastically.
This was vital in understanding the MRI signal equation and is thoroughly used in the continual development of MRI today.
$\endgroup$ $\begingroup$You can write:
$\sin x = \operatorname{Im}(e^{ix})$, $\cos x = \operatorname{Re}(e^{ix})$.
which is often useful. More than that, consider:
$e^{i\theta} = \cos \theta + i \sin \theta$
$e^{-i\theta} = \cos \theta - i\sin \theta$
Where the second equation comes from $\cos$ being even and $\sin$ being odd.
Subtract or add the equations together and you can express $\sin$ and $\cos$ completely in terms of the exponential function! And once you have those you can do the other 4 trig functions.
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