I think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.

The best explanation I've seen, which 3blue1brown probably avoided because it requires calculus, is that exp is the unique function whose value at 0 is 1 and whose derivative is itself. Then by the chain rule the derivative of exp(ix) is i*exp(ix), meaning that the direction of motion is perpendicular to the current position vector. Which naturally leads to circular motion because you're always staying the same distance from the origin.

With that definition it's easy to derive the Taylor series expansion (every derivative at 0 is 1), and you can think of Euler's formula not as telling you how to evaluate exp(ix) (it's already perfectly well defined), but as an introduction of cos and sin as shorthand for its real and imaginary parts.