What does $\tan x = 1$ mean? 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?
Thanks

Solving the equation $\tan x=1$ means finding every angle $x$ whose tangent is $1$. You can start with $\frac{\pi}4$, the principal arctangent of $1$; the tangent has period $\pi$, so (as you said) you need to add integer multiples of $\pi$ to that to get the entire set of solutions:
$$\left\{\frac{\pi}4+n\pi:n\in\Bbb Z\right\}\;.$$
If you’re asked to solve $\sin x=2$, you can safely say that the set of solutions is empty ($\varnothing$), since $|\sin x|\le 1$ for all $x\in\Bbb R$.
In neither case is it actually necessary to think in terms of graphs, though it’s certainly possible to do so: the set of solutions to $\tan x=1$ is indeed the set of $x\in\Bbb R$ where the graphs of $y=\tan x$ and $y=1$ intersect, and similarly for $\sin x=1$.
