not sure how a function of e became the hypotenuse of a trigonometric triangle within a circle in the complex plane. So far, the reasoning seems circular... still computing ...
(hints of which websites to check would be much APPRECIATED) 

e^ix=cosx + isinx

When x= i, then
e^(i•i) = e

If cos(i) is imaginary, then i•sin(i) can't be.

If cos(i) isn't imaginary, then i•sin(i) must be.

Either way an imaginary number plus a non-imaginary number can't be a real number such as e.

e^(ix) = cos(x) + i•sin(x) cannot be correct for all numbers including i.