Remember that $\arcsin$ is supposed to be the inverse function of $\sin$ (or at least, of a "restricted sine function"; that's the choice of 'principal value').
The way that $\arcsin$ works is supposed to be: you plug in the value somebody else got out of the sine function, and $\arcsin$ will tell you what number was put into the sine function to get that value. It's like a "reverse telephone directory": you look up the phone number and find out the person it belongs to, instead of the usual way of looking up the person and finding their phone number.
But that means that the only things that you can put into the $\arcsin$ functions are real numbers that actually come out of the sine function (the only numbers you can look up in a reverse telephone directory are telephone numbers, so you can't look up "000-0000").
What are the numbers that can come out of the sine function? Every number between $-1$ and $1$ (inclusively), but only those numbers: the sine function will never give a result that is greater than $1$ or smaller than $-1$. That means that the numbers you can plug into $\arcsin$ are only the numbers that come out of the sine function: the numbers between $-1$ and $1$.
The same thing is true for $\arccos$: you can only put into $\arccos$ numbers that may come out of the cosine function, and the only numbers that may come out of the cosine function are the numbers on $[-1,1]$.
However when we come to the $\arctan$ function, things are different: what are the numbers that may come out of the tangent function? Every real number! Every real numbers is the tangent of some angle, so now we can put into $\arctan$ any real number, because any real number is, potentially (and in actuality) the result of applying the tangent function.
Note. Your final paragraph seems to be confusing the "trigonometric tangent function" with the tangent line to the graph of a function. The "trigonometric tangent function" is the function defined by
$$\tan(x) = \frac{\sin(x)}{\cos(x)}.$$
The "tangent line to the graph of $f(x)$ at $x=a$" is a straight line that has certain properties (it goes through $(a,f(a))$, and is the straight line that "best approximates" the graph of $y=f(x)$ near the point $(a,f(a))$). The derivative, which is a key concept in calculus, is the slope of the tangent-line-to-the-graph-of-$f(x)$ (which is defined as a limit of a certain ratio), not to the trigonometric tangent function which is what $\arctan(x)$ is related to.
Congratulations! You've stumbled in to a very interesting question!
In higher mathematics, we often notice that some things which are really easy to talk about but difficult to express rigorously have a property which is really easy to express rigorously but something that we probably wouldn't have thought of to begin with.
The trig functions are one of these things. With (a lot of) effort, you can show that
$$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \cdots $$
where the patterns of increasing the powers of $x$ by $2$, and switching between $+$ and $-$ signs continues forever. (The denominators also have a pattern: take the power that $x$ is raised to in the term and multiply it by all of the smaller numbers down to $1$; that is the number in the denominator). Note that you have to use radians for this exact formula to work; of course you could come up with one for degrees as well.
When you start realizing that circles are actually quite tricky objects to define, formulas like that one start to look more appealing. I have had multiple mathematics textbooks take this infinitely long expression as the definition of the sine function. (It turns out to be the same thing as the circle definition, but… well, circles get complicated.)
Of course, we can't sit around multiply and add for the rest of our lives just to compute sin $1$, but we can just cut off the operations after a couple terms. If you go out to the $x^7$ term, you can guarantee that your answer is accurate to at least 3 decimal places as long as you use angles between $-\frac{\pi}{2}$ and $\frac\pi 2$. (These are the only angles you really need, if you get rid of multiples of $\pi$ properly.)
The cosine formula, in case you are interested, is similar:
$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}+ \frac{x^8}{40320}-\cdots$$
The internet has formulas for the other trig functions, but you can always just combine these.
As copper.hat says, there are also these large books where people did the calculations once and wrote them down so that nobody would have to do them again. Of course, these were made long before computers existed; nobody makes them anymore! But somebody from your parents' or grandparents' generation probably still has one sitting in their house.
Best Answer
If you complete the diagram with all the right triangles, you would notice all of them are actually similar, meaning pairs of corresponding sides produce equivalent ratios. I’m not exactly sure how to explain this considering there aren’t any points in your diagram, so here’s a link to the proof I used.
Trigonometric Ratios on a Unit Circle