Question Number 209519 by depressiveshrek last updated on 12/Jul/24
$$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:{n}\in\mathbb{N}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{sin}\left({n}\right)\in\mathbb{Q}\:? \\ $$
Commented by Rasheed.Sindhi last updated on 12/Jul/24
$${n}\:{in}\:{degrees}? \\ $$
Commented by depressiveshrek last updated on 12/Jul/24
$$\mathrm{No}\:\mathrm{that}\:\mathrm{would}\:\mathrm{be}\:\mathrm{too}\:\mathrm{easy} \\ $$
Commented by mr W last updated on 12/Jul/24
$${no}!\:{there}\:{is}\:{even}\:{no}\:{rational}\:{number} \\ $$$$\boldsymbol{{n}},\:{except}\:{zero},\:{such}\:{that}\:\mathrm{sin}\:\left(\boldsymbol{{n}}\right)\:{is} \\ $$$${rational}. \\ $$$${actually}\:{the}\:{only}\:{rational}\:{values}\:{of} \\ $$$$\mathrm{sin}\:\left({x}\right)\:{are}:\:\mathrm{0},\:\pm\frac{\mathrm{1}}{\mathrm{2}},\:\pm\mathrm{1}. \\ $$$$\Rightarrow{see}\:{also}\:{Niven}'{s}\:{theorem} \\ $$