The Euler Equation: What does "e" have to do with sine & cosine?

First, thank you to all the smart people who responded to my last
Euler Equation post...

I'm trying to understand this equation from an intuitive perspective.
For example, what does the number "e" have to do with sine and cosine?

I've come up with a few hints about this, and I've outlined them
briefly below. Maybe someone out there can take this further and show
why "e" is related to sine and cosine.

j^0 = 1
j^1 = j
j^2 = -1
j^3 = -j ... and so on

This shows us that we rotate counter-clockwise in the complex plane as
we take j to successively higher powers.

Since j^0 = 1, and j^1 = j, we might infer that j^(1/2) is half way
between (i.e. at an angle of pi/4 radians), and has a real part and an
imaginary part:

j^(1/2) = cos(pi/4) + jsin(pi/4)

So now we can see how j is related to sine and cosine. I'm wondering if
anyone can now show how "e" is related to sine and cosine. I'm not
really looking for a proof, but rather an explanation.

Thanks in advance.

.

Relevant Pages