system
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human
I feel bad for not knowing this as an engineering junior and I've kinda been blindly doing it for like 2 years now so I never thought to ask about this simple trig concept...still doesnt make sense.

Let's say I have a right triangle and where we have the adjacent and opposite sides to an angle theta which we don't know. I'll make up some numbers for how long they are. 

So we have

Adjacent = 1

Opposite = 2

Theta = ????

So im like cool

tan(theta) = 2/1

then to get what theta I take the inverse tangent of 2. I know that there are other ways to solve something like this using law of sins or law of cosines...but if simply wanted to use tangent.

tan\^-1(2) = Theta and throw it in a calculator and boom im done. But can somebody explain steps why that works.  Because I thought that a trig value of some variable leads to a ratio and that within a trig expression that there has to be some sort of variable like cos(x), sin(theta).... Are you just dividing by the tangent on both sides and then you have something like Theta = 2/ tan. I just dont get why that works.
gpt
It seems you confuse multiplicative inverse with inverse function. The multiplicative inverse x=a^(−1) is the number that solves xa=1, while the inverse function is used to solve f(x)=a: when you write x=f^(−1)(a), then x is by definition of f^(−1) the solution of f(x)=a. In your case, it is f=tan, a=2, x=θ.