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Question Number 71448 by aliesam last updated on 15/Oct/19

find the domain    f(x)=(1/(sin(sin(x))))

$${find}\:{the}\:{domain} \\ $$$$ \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{sin}\left({sin}\left({x}\right)\right)} \\ $$

Commented by kaivan.ahmadi last updated on 15/Oct/19

sin(sinx)=0⇒sinx=kπ  ;k∈Z  but since −1≤sinx≤1 so −1≤kπ≤1⇒k=0⇒  sinx=0⇒x=kπ  ;k∈Z  ⇒D_f =R−{kπ ∣k∈Z}

$${sin}\left({sinx}\right)=\mathrm{0}\Rightarrow{sinx}={k}\pi\:\:;{k}\in\mathbb{Z} \\ $$$${but}\:{since}\:−\mathrm{1}\leqslant{sinx}\leqslant\mathrm{1}\:{so}\:−\mathrm{1}\leqslant{k}\pi\leqslant\mathrm{1}\Rightarrow{k}=\mathrm{0}\Rightarrow \\ $$$${sinx}=\mathrm{0}\Rightarrow{x}={k}\pi\:\:;{k}\in\mathbb{Z} \\ $$$$\Rightarrow{D}_{{f}} =\mathbb{R}−\left\{{k}\pi\:\mid{k}\in\mathbb{Z}\right\} \\ $$

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