From the modern perspective, Euclidean geometry is the study of the topological space
$$\Bbb{R}^n = \{ x_1, \ldots, x_n \; \mid \; x_i \in \Bbb{R} \}$$
together with the bilinear form
$$\langle \cdot, \cdot \rangle: \Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}_{\ge 0}$$
defined by
$$\langle \vec{x}, \vec{y} \rangle = \big\langle (x_1, \ldots, x_n), (y_1, \ldots, y_n) \big\rangle = x_1y_1 + \cdots + x_ny_n$$
This "dot" product defines distances and angles via
$$\| \vec{x} \| = \sqrt{\langle \vec{x}, \vec{x} \rangle} = \sqrt{x_1^2 + \cdots + x_n^2}$$
and
$$\cos \theta = \frac{\langle \vec{x}, \vec{y} \rangle}{\sqrt{\langle \vec{x}, \vec{x} \rangle \langle \vec{y}, \vec{y} \rangle}} = \frac{\langle \vec{x}, \vec{y} \rangle}{\| \vec{x} \| \| \vec{y} \|}$$
From the perspective of transformations (a decidedly modern take), certain functions
$$\phi: \Bbb{R}^n \to \Bbb{R}^n$$
are more interesting: conformal maps preserve angles (these are also called similarity transformations):
$$\langle \phi(\vec{x}), \phi(\vec{y}) \rangle = \langle \vec{x}, \vec{y} \rangle \qquad \text{for all } \vec{x}, \vec{y} \in \Bbb{R}^n$$
and certain conformal maps, called isometries, preserve distances as well (these are also called congruence transformations):
$$\| \phi(\vec{x}) \| = \| \vec{x} \| \qquad \text{for all } \vec{x} \in \Bbb{R}^n$$
In the plane $(n = 2)$ there is a nice characterization of all isometries as translations, rotations, reflections, and glide reflections. (Throw in dilations to get conformal maps.)
By equipping $\Bbb{R}^2$ with the imaginary unit $i$ $(i^2 = -1)$, it becomes the complex plane, and all of these isometries can be written as rational functions $\phi: \Bbb{C} \to \Bbb{C}$. (Note that the complex structure is really essential to define rotations.) EDIT: You need complex conjugation, too, in order to write down any orientation-reversing map, such as a reflection.
In that sense, the complex numbers are the perfect algebraic object to capture the Euclidean structure of $\Bbb{R}^2$ and express these maps in elegant formulas.
Best Answer
The fact that $\exp(i(\theta_1+\theta_2))=\exp(i\theta_1)\exp(i\theta_2)$ immediately leads to many trigonometric formulas, including the most basic of $\cos(\theta_1+\theta_2)$ and $\sin(\theta_1+\theta_2)$. Other good examples are $\sin 3\theta,\,\sin 4\theta,$ etc.
The easiest way to find the coordinates of a right polygon with $n$ vertexes is to find $n$ $n$th roots of 1.