The formal definition of “angle”
This is a great question. And the first part of the answer to "How do you define angle" is "Lots of different ways." And when you define it a new way, you have some responsibility for showing that this new way is consistent with the old ways.
The next thing to realize is that there's a difference between an angle and the measure of that angle. In classical geometry, an "angle" is a pair of rays with a common starting point (or a pair of intersecting lines --- depends on your textbook), while the measure of an angle...well, that's actually not such a classical notion: the Greek geometers were more inclined to talk about congruent angles, and leave "measure" to "mere" tradesmen, etc. But if we look at a modern view of classical geometry (e.g., Hilbert's axioms) then the measure of an angle is a number --- clearly a different kind of entity from a "pair of rays".
If you're going to talk about periodic motion, then it's useful to think of a ray starting out pointing along the $x$-axis and then rotating counterclockwise so that it makes ever-larger angles (by which I mean "angles of ever-larger measure") with the $x$-axis. But when you reach a measure of $2\pi$ radians or $360$ degrees, you're back to the same "angle" consisting of two copies of the positive $x$-axis. From an analysis point of view, it's nice to think of the "angle measure" as continuing to increase, so we talk about an angle of, say, $395$ degrees.
Right at that moment, the analyst/physics person/tradesman has diverged from the geometer. For smallish angles, they agree; for large ones they disagree. But it's no big deal -- people can usually figure out what's going on from context.
If we think of our angle as being situated at the origin (now I'm a coordinate geometer rather than a Euclidean one!), the two rays subtend an arc of the unit circle at the origin. And some folks (including me!) might call that arc an "angle". So to those folks --- I'll call them "measure theory" people --- an angle is an arc of the unit circle. And the measure of the angle is simply the measure of the arc ... which has to be defined via notions of integration, etc. It's very natural for such a person to say "oh...and I'd like to say that an 'angle' is not just a single arc, but any (measurable) subset of the unit circle." Once you say that, "additivity" of angles follows from additivity of measures. (I don't mean to say this is easy! There's lots of stuff to say about rotationally-invariant measures, etc.)
That measure-theory generalization now lets you define things like "solid angles" in 3-space: a solid angle is just a (measurable) subset of the unit sphere. But the measure-theory approach also loses something: there's no longer such a thing as a "clockwise" and a "counterclockwise" angle, at least not without a good deal of dancing around.
To return to your question about the law of cosines: there's a kind of nice approach to relating cosines to geometry: you show that there are a pair of functions $s$ and $c$ (for "sine" and "cosine", of course) defined on the real line that satisfy three or four basic properties, like $c(0) = 1$, and $c(x-y) = c(x)c(y) + s(x) s(y)$. You do this in a few steps: first, you show that for rational multiples of $\pi$ (which appears in one of the properties), the values of $c$ and $s$ are uniquely determined (i.e., you show that there's at most one such pair of functions). Letting $P$ denote these rational-multiples-of-$\pi$, you then show that the functions $c$ and $s$ are periodic of period $2\pi$ (on $P$), and then use some classical geometry to explicitly show that they can be defined on the set of all geometric angles with geometric angle measure in the set $P' = P \cap [0, 2\pi)$. Then you show that on $P$, these functions are continuous, and apply the theorem that every continuous function on a dense subset of a metric space admits a unique continuous extension to the metric space itself, thus allowing you define $c$ and $s$ on all of $\Bbb R$. Now this pair of functions has exactly the properties that the geometric definition would assign to the "cosine" and "sine" [a fact that comes up while proving that $c$ and $s$ can be defined on geometric angles with measures in the set $P'$]. So you've got sine and cosine functions with all the properties you need, but no calculus involved. (I believe that this whole development is carried out in Apostol's Calculus book.) Finally, you can look at the dot-product definition of angles as a way of defining a new function --- let's call it csine: $$ csine(\theta) = \frac{v \cdot w}{\|v\| ~\|w\|} $$ that depends on the angle $\theta$ between two vectors $v$ and $w$.
Now you show (lots of linear algebra and geometry here) that this function satisfies the very "properties" I mentioned earlier --- the essential one being closely tied to the law of cosines), and that it must therefore actually be the same as the cosine function we defined a paragraph or two earlier.
I want to mention that this connection of all these things took me years to learn. I sort of knew a bunch of them, but I don't suppose it was until a decade after I got my Ph.D. that I could have put them all together into a coherent thread, a thread that I've only barely summarized here.
I have found a satisfactory answer in the context of Geometric Algebra. I have just written it at
https://whatihavelearnedajpan.blogspot.com/2020/07/angles-from-geometric-algebra.html
I have tried to modify it, to paste here, but is too much time for me.
By the way, this definition let you connect the idea of "exponential" and a definition for the $\pi$ number.
To give a summary: In GA, bivectors represent 2-dimensional directions. Any simple bivector can be written as $$ \theta e_1e_2 $$ with $\theta \in \mathbb{R}$ and $\{e_i\}$ unitary and orthogonal vectors. The unitary vectors specify the direction itself, and $\theta$ is the size (analogous to the length of a vector, that is a 1-dimensional direction).
Given two vectors $a$ and $b$ (that we take to be unitary, wlog) we say that the simple bivector $\theta e_1e_2$ is the angle formed by them if $$ ab=e^{\theta e_1 e_2} $$ where we take as a definition $$ e^A=1+A+A^2 /2+\cdots=\lim_N (1+\frac{A}{N})^N $$
The exponential could be interpreted as an infinite process, as you can see in the gif below:
since $$ b=a e^{\theta e_1 e_2}\cong a(1+\frac{\theta e_1 e_2}{N})^N=a+\frac{\theta}{N}a_{\perp}+\cdots $$