Question Number 209519 by depressiveshrek last updated on 12/Jul/24
![Is there any n∈N such that sin(n)∈Q ?](https://www.tinkutara.com/question/Q209519.png)
$$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:{n}\in\mathbb{N}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{sin}\left({n}\right)\in\mathbb{Q}\:? \\ $$
Commented by Rasheed.Sindhi last updated on 12/Jul/24
![n in degrees?](https://www.tinkutara.com/question/Q209523.png)
$${n}\:{in}\:{degrees}? \\ $$
Commented by depressiveshrek last updated on 12/Jul/24
![No that would be too easy](https://www.tinkutara.com/question/Q209524.png)
$$\mathrm{No}\:\mathrm{that}\:\mathrm{would}\:\mathrm{be}\:\mathrm{too}\:\mathrm{easy} \\ $$
Commented by mr W last updated on 12/Jul/24
![no! there is even no rational number n, except zero, such that sin (n) is rational. actually the only rational values of sin (x) are: 0, ±(1/2), ±1. ⇒see also Niven′s theorem](https://www.tinkutara.com/question/Q209527.png)
$${no}!\:{there}\:{is}\:{even}\:{no}\:{rational}\:{number} \\ $$$$\boldsymbol{{n}},\:{except}\:{zero},\:{such}\:{that}\:\mathrm{sin}\:\left(\boldsymbol{{n}}\right)\:{is} \\ $$$${rational}. \\ $$$${actually}\:{the}\:{only}\:{rational}\:{values}\:{of} \\ $$$$\mathrm{sin}\:\left({x}\right)\:{are}:\:\mathrm{0},\:\pm\frac{\mathrm{1}}{\mathrm{2}},\:\pm\mathrm{1}. \\ $$$$\Rightarrow{see}\:{also}\:{Niven}'{s}\:{theorem} \\ $$