The domain of $\arcsin(\arccos(x))$
This is an exam-style question where I must find the implied domain of $\arcsin(\arccos(x))$ and express the answer in terms of cosine.
I have $-1 \leq \arccos(x) \leq 1$
The answer is $\cos(1) \le x \le 1$, but I'm not sure how to get to this part, online calculators do not explain this step at all. Any help is appreciated!
$\endgroup$1 Answer
$\begingroup$The domain consists of those numbers $x$ which:
- belong to the domain of $\arccos$;
- are such that $\arccos x$ belongs to the domain of $\arcsin$.
The domain of $\arccos$ is $[-1,1]$ and its range is $[0,\pi]$. The domain of $\arcsin$ is $[-1,1]$. So, the domain is the set of those $x\in[0,\pi]$ such that $\arccos x\in[-1,1]$. So, the domain is$$\{x\in[-1,1]\mid\arccos(x)\leqslant1\},$$which is precisely $\bigl[\cos(1),1\bigr]$.
$\endgroup$ 1
