My teacher teacher told me that for a general angle $x$, $\cos^{-1}({\cos{x})}$ does't represent $x$ but different straight lines depending upon the intervals in which it lies. For ex:
$$\cos^{-1}{\cos{x}}=$$
$$x,0\leq x \leq \pi \\ 2\pi-x,\pi\leq x \leq 2\pi\\…$$
making the graph look like :-
From wolfram alpha
He told us that if we have to find the value of $\cos^{-1}({\cos{x})}$ for a particular $x$ we will have to first find the range in which $x$ lies and then judge with the help of graph but I wondered if there is a direct formula for that. I tried with $\tan^-1({\tan{x}})$ and got it as :-
from wolfram alpha
I even verified this with wolfram alpha and got it right but the problem with $\cos$ is that when I try to solve it similarly like I did with the $\tan$ one, and get the interval in which $n$ lies, the extremities of the interval differ by $0.5$ because of which for some values their floor and ceiling match but for some values there isn't an integer value lying in that interval like this :- from wolfram alpha
so what to do in that case and what does no value of $n$ lying in the interval signify?
Thanks for help 🙂
Best Answer
By definition we have that for $x\in[0,2\pi]$
and this is periodic with period $T=2\pi$.
Thus it is a kind of triangle function and we always need to divide into two parts dependind upon the range in which x lies.