Derivation

We start with complex numbers, of the form

$$z = x + \imath y$$

which has an Argand diagram which looks like

 

Figure 1: Argand Diagram

We know that $x = r \cos \theta$ and $y \ r \sin \theta$ by elementary trigonomentry, and that the inverse relations obtain

$$r = \sqrt{x^2+y^2}\ \ \;\ \ \theta = \arctan \frac{y}{x}$$

If we expand $e^{\imath \theta}$ in a Taylor Series one has

$$e^{\imath \theta} = 1 + \imath \theta + \frac{1}{2!} (\imath... ...!} (\imath \theta)^3 + \frac{1}{4!} (\imath \theta)^4 + \cdots$$

which results in

$$e^{\imath \theta} = 1 + \imath \theta - \frac{1}{2!} (\theta... ...c{1}{3!} \imath (\theta)^3 + \frac{1}{4!} ( \theta)^4 + \cdots$$

where we form two alternating series, one imaginary, the other real. The $\sqrt{-1}$ can be factored from the imaginary series, so that we finally obtain (recognizing the Taylor expansions of sine and cosine)

$$e^{\imath \theta} = \cos \theta + \imath \sin \theta$$

which is DeMoivre's (Euler's) Theorem.