I'm reviewing math material for the EIT exam, I'm going over math concepts that should be pretty basic but I feel like there are gaps in my understanding. I understand how we can use rectangular coordinates and complex numbers to find a point on the complex plane. It would follow logically from trig that the rectangular coordinates z=x+iy that z=r(cos(θ)+isin(θ)) being that x=cos(θ) and y=sin(θ) I also know the definition z=reiθ however why is this definition true? Can anyone explain it to me? In addition to this it can easily be assumed that since z=reiθ and z=r(cos(θ)+isin(θ)) that eiθ=(cos(θ)+isin(θ)) giving us Euler's formula. My next question comes from exactly that. How is it that trig functions relate to the exponential function is this just something to accept or is there more underlying causes? Did Euler come up with his equation from the addition of the Maclaurin series of e, sin, and cos; or did he figure this out some other way?