4. A very useful formula, used in the complex-domain analysis of linear systems, is Euler's

formula which relates the function eix (here, i = √-1) to sinusoids, according to:

eix = cos(x) + i sin(x)

Employ the power series representations of ex, cos(x) and sin(x) to derive Euler's formula.

(Side note: For the special case of x = , Euler's formula reduces to Euler's identity: ein +

1 = 0. It has been said that this is the most "beautiful" mathematical expression relating

e, I, I, 1, 0.)