Understanding imaginary exponents is certainly nice and worth achieving. However, to understand the relationship between trigonometry and complex numbers, you do not need to understand imaginary exponents.
The imaginary exponents only arise if you feel the need to phrase things in terms of e, rather than (what I called above) R. But there’s no real need to do this.
I really like the above articles’ explanation of circular motion in the complex plane. There is definitely a eureka moment in there with seeing how the properties of i cause things to go in a circle, and cos(x) + isin(x) makes perfect sense when explained visually. (It’s exactly the same as the unit circle, except the Y coordinate is multiplied by i to make ti work on the complex plane.)
That’s right. Circular motion in the complex plane is a fantastic thing to understand, and cos(x) + i * sin(x) is precisely what it is. I’m just saying, you can say “R^x = cos(x) + i * sin(x)” instead of “e^(ix) = cos(x) + i * sin(x)” and it’s all the same; you don’t have to worry about imaginary exponents if you don’t want to, and you don’t even have to think about e when it isn’t what you care about.
That having been said, understanding how R = e^i is fine and dandy as well. It just isn’t a necessary part of the connection between trig and complex numbers that you’re interested in.
In other words: once you already understand that i is 90 degree rotation (i.e., a quarter of a full turn) and how multiplying combines rotations, you’re set.
i^2 = i * i = the cumulative result of 2 quarter turns = 180 degree rotation.
i^3 = i * i * i = cumulative result of 3 quarter turns = 270 degree rotation.
i^(1/2) = 1/2 of a quarter turn = 45 degree rotation.
i^(3.3) = 3.3 quarter turns = 297 degree rotation
In this vein, exponentiation with the imaginary base i, but real exponents, is already enough to describe arbitrary rotation. You don’t have to worry about imaginary exponents to see the connection between complex numbers and rotation; all that matters are suitable bases.
It happens to be the case that ln(i) = 1/4 * 2πi = iπ/2, so that i = e^(iπ/2), giving us an imaginary exponent. But you don’t have to understand why ln(i) = 1/4 * 2πi to understand the connection between complex numbers and rotation.
That having been said, if you want to understand why ln(i) = 1/4 * 2πi, I can try to help you with this (it’s based on the way ln(x) is defined from the rate of growth of the function x^t). But understanding this, while useful and worth achieving, is a separate project beyond simply seeing the relationship between complex numbers and rotation/trigonometry.