Euler's Formula

On the circumference of a unit circle, a point rotating at a constant speed can have its angle considered linear, written as $\omega t$. According to trigonometric functions, the point's horizontal coordinate is $\cos(\omega t)$, and its vertical coordinate is $\sin(\omega t)$. Therefore, in the complex plane, the position of this point can be represented as follows:

$$\cos(\omega t) + j \sin(\omega t)$$

According to Euler's formula,

$$e^{j\omega t} = \cos(\omega t) + j \sin(\omega t)$$

Thus, the position of the point can also be written as $e^{j\omega t}$; meaning $e^{j\omega t}$ represents a signal point rotating at a constant angular velocity of $\omega$.

If we replace $\omega$ with $-\omega$ in Euler's formula and use it together with the above, we can perform some simple calculations to arrive at the following formulas:

$$\cos(\omega t) = \frac{e^{j\omega t}+e^{-j\omega t}}{2}$$

$$\sin(\omega t) = \frac{e^{j\omega t}-e^{-j\omega t}}{2j}$$

In other words, both $\cos(\omega t)$ and $\sin(\omega t)$ can be seen as the composition of two rotating waves at a constant speed, one being $\frac12 e^{j\omega t}$ and the other being $\frac12 e^{-j\omega t}$.

If we can determine the rates of rotation of the two waves through sampling, we can determine the angular velocity of $\cos(\omega t)$ and $\sin(\omega t)$.