Re: 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.


The fact that you used "j" instead of "i" proves that you are an engineer and thus not capable of understanding what "e" has to do with sine and cosine! :)

The simple fact is that "sin x", "cos x" and "e^x" are PARTS of the function e^z in the complex numbers.
.

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