Question

Where did the exponential series come from?

1 + x + x^2/2! + x^3/3!...

Where did that number come from?

and how is it used to get the trigonometric series?

Answer 1

From Taylor (Maclaurin series if starting at x =0) series
f(x)=f(0) + f'(0)(x/1!) + f""(0)x^2/2!)....
here f(x) = e^x
f(0) = 1
the derivatives of e^x are all e^x which is 1 at x = 0
so
e^x = 1 + 1(x/1!) +1(x^2/2! etc

Answer 2

note
sin x = (e^ix - e^-ix)/2
cos x = (e^ix + e^-ix)/2

Answer 3

sin x = (e^ix - e^-ix)/2i
forgot i
cos x = (e^ix + e^-ix)/2