I can solve for $x$ by taking the arctangent of both sides but I'm not understanding what the equation means. Does the equation represent the interesection between $y = \tan x$ and $y =1$? Likewise is $\sin x = 2$ said to be undefined as the two curves do not intersect for any real $x$ (they don't cut each other?). I feel that I am missing a key concept in trigonometric equations.
Also to find other solutions of $\tan x = 1$ do I just subtract or add $\pi$ to the principle value?