Intuitive understanding of why trigonometric functions relate to both triangles and the complex exponential The trigonometric functions sine and cosine can be defined in terms of the complex exponential:
$$\sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix})$$
$$\cos(x) = \frac{1}{2}(e^{ix} + e^{ix})$$
And can also be defined in terms of the lengths of the sides of a triangle where $x$ now represents the angle between the hypotenuse and the adjacent sides $H$, $A$, and $O$ represent the lengths of the hypotenuse, adjacent, and opposite sides respectively:
$$\sin(x) = \frac{O}{H}$$
$$\cos(x) = \frac{A}{H}$$
I find it utterly bizarre that something as seemingly unrelated as the dimensions of a triangle relates to exponentiation and the root of a negative.
Mathematically, how is this possible?
Intuitively, why is this possible?

$$e^{ix}=\cos x + i \sin x$$
is intuitively a circle of radius $|e^{ix}|=1$ in the complex plane with the radius, the real part, and the complex part forming a right-angled triangle.

