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I was taught that Eulers formula defined complex exponents?

If we used turns for cos and sin we could redefine what e^ix means so it works without radians. From the other answer I guess this is completely wrong...

(I do understand it is nuts to redefine, i'm just interested as a theoretical thought)

Now, how is Eulers formula is deduced? How did we figure out what e^ix means?

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One way to understand where the formulas come from is the power series of e^x, remembering that that function is (can be) defined as the function whose derivative is itself. Sin and cos are functions whose second derivative is -sin and -cos respectively. If you plug in ix to the power series for e^x, the complex exponential comes right out.

There are a couple other "paths" to this result, and the choice we have is by far the most elegant.

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