How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite (Galois) field. I've read the Wikipedia article but I'm having trouble understanding what sorts of angles and numbers are representable in finite fields.

Here is what I do understand:

Starting with the 2D Cartesian plane with coordinates x, y, we can represent discrete angles that are multiples of $90^\circ = \frac{\pi}{2}$. These are the fourth roots of unity $x = \cos{\frac{2k\pi}{4}}$ and $y = \sin{\frac{2k\pi}{4}}$ or alternatively:
$z = \cos{\frac{k\pi}{2}} + i\sin{\frac{k\pi}{2}}$, where $k$ is a positive integer less than $4$. These numbers can be represented solely with the integers. If we want to add discrete angles that are multiples of $30^\circ = \frac{2\pi}{12}$, we need a quadratic extension of the integers so that we have quadratic (algebraic) integers of the form $a + b\sqrt 3$. This allows us to represent the twelfth roots of unity as x and y coordinates. If we wish to double the number of angles to $15^\circ = \frac{2\pi}{24}$ multiples, we must extend our field again, forming a tower of quadratic extensions with numbers of the form $(a + b\sqrt 3) + (c + d\sqrt 3)\sqrt 2$. Numbers of this form allow us to represent the $24^{th}$ roots of unity.

How does this work in a finite field? Can I choose a finite field such that I can exactly represent the $n^{th}$ roots of unity in a manner analogous to the above? I'm particularly interested in constructable numbers, which feature only quadratic extensions (and multiquadratic extensions like $\sqrt{5 + \sqrt 5}$). In particular this means that $n$ is restricted to having factors of 2 and Fermat primes. I restricted myself to powers of $2$ and Fermat prime $3$ in my example above. Both $12$ and $24$ have factors of only $2$ and $3$.

- Edit -

To try to clarify what I'm struggling with. I do not see how to find or use a finite field that has been extended twice or more (e.g. angles of $\frac{\pi}{12}$ as described above), as the relationship to the complex plane in a finite field setting seems to blur as the tower of extensions grows.

This is a new subject for me, so I'd really appreciate an example or two to go along with any explanations.