A complex exponential traces a circle, with its velocity always turned a quarter-turn.
Expand exp(iθ) in its absolutely convergent power series. The even powers of i alternate between 1 and −1 and sum to cos θ. The odd powers alternate between i and −i and sum to i sin θ. Absolute convergence permits separating these sums for every real θ.
The imaginary unit is part of the coordinate system; the unit circle still has ordinary real x and y coordinates.
The short red tangent arrow points in the direction iz. Multiplying by i rotates a vector 90° counterclockwise. This picture illustrates the identity; its power-series proof appears in the explanation.