Proof of Eulers formula

As we know Euler’s fomula is very ubiquitous in mathematics, physics, and engineering. The fomula is as follows:

\[e^{i\theta}=\cos(\theta)+i\sin(\theta)\]

I never though how to prove this formula until seeing a simple proof with an interesting story behind in the book “Linear Algebra and Its Applications”, Gilbert Strang. One day there was a letter came to MIT from a prisoner in New York, asking if Euler’s fomula was true? It is really astonishing. Three key functions of mathematics come together in such a graceful way.

The proof of the Euler’s fomulas, in fact is quite simple using the Tayler series.

\[e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \frac{(i\theta)^5}{5!} ...\]

Because $i^2=-1$,

Real$( e^{i\theta})= 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} + … = \cos(\theta) $, and

Img$( e^{i\theta})= i(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} + …) = i\sin(\theta).$

Thus the formula is correct.