The lecture in the sequences and series portion of calculus where the prof took the Taylor series of e^x and then substituted x=i*theta into the resulting polyn

The lecture in the sequences and series portion of calculus where the prof took the Taylor series of e^x and then substituted x=i*theta into the resulting polynomial and all of the terms magically divided themselves into the Taylor series’s for sin and cos was a jaw dropping moment for me. I was in my second year of EE, and it completely changed my relationship with a ton of practical material for me, eg the Fourier transform. On the other hand, I also took vector calculus that year but it wasn’t until the following year when I took the E&M course (electric and magnetic fields) where I truly appreciated surface integrals, path integrals, etc. One of the biggest Aha! moments there was really appreciating how properly structuring the problem could dramatically simplify the computations. Eg if you can make it symmetric you can often cause integrals to disappear implicitly without having to compute them at all, or you can take a dramatically simpler path through a vector field and end up at the same place without a ton of complex computation. Absolutely beautiful :D