Unit circle of the complex plane

Unit circle of the complex plane

the diagram

Vectors in the complex plane from the origin to a point on the unit circle have a special notation in terms of a complex exponential.
Such a vector, v, has the following components:

(1)

and

(2)

By virtue of spanning from the origin to the unit circle, the length of the hypotenuse, h, produced by the vector is 1. So, the definitions of sine and cosine

(3)

and

(4)

in this case reduce to

(5)

and

(6)

Placing these expressions for a and o into eqns. 1 and 2 yields

(7)

and

(8)

Now reconstructing the vector from these components, we have

(9)

Euler's theorem provides a concise way of representing the right side of this equation in terms of a complex exponential, given by

(10)

So, by combining eqns. 9 and 10 by their right sides, the vector v can therefore be represented by

(11)