This apparently obvious translation between complex math and trigonometric functions is in fact a great conceptual leap. Long before they were drawn as points on a plane, complex numbers were conceived and manipulated as pure numbers. Mathematicians no more visualized complex arithmetic as rotations between axes any more than they would have drawn a number line to add two ordinary numbers.
Nor was trigonometry on hand to point the way to this interpretation. The subject was considered a mathematically interesting sideline of geometry, studied by surveyors and astronomers, and was not part of the mainstream of mathematics. The sine and cosine functions were known and used, but there was little to suggest their involvement in the math of pure numbers.
So we've made a great leap of mathematical faith with the analogy between complex and trigonometric math. The complex numbers can be conceptually illustrated by number lines at right angles because of the properties of i; but to treat this literally as a geometric right angle and apply trigonometry to it is a far-sighted gamble with no guarantee of mathematical validity. In this case, though, the correspondence has held soundly right to the core. A hidden truth waiting to be realized, it's one of the great discoveries in math.