Question Number 153593 by tabata last updated on 08/Sep/21
$$\boldsymbol{{Solve}}\::\:\left(\boldsymbol{{sin}}\left(\mathrm{2}\boldsymbol{{x}}\right)\right)!\:=\:\mathrm{2} \\ $$
Answered by puissant last updated on 08/Sep/21
$$\left({sin}\left(\mathrm{2}{x}\right)\right)!=\mathrm{2}!\:\Rightarrow\:{sin}\left(\mathrm{2}{x}\right)=\mathrm{2} \\ $$$$\Rightarrow\:\frac{{e}^{\mathrm{2}{ix}} −{e}^{−\mathrm{2}{ix}} }{\mathrm{2}{i}}=\mathrm{2} \\ $$$$\Rightarrow\:{e}^{\mathrm{4}{ix}} −\mathrm{1}=\mathrm{4}{ie}^{\mathrm{2}{ix}} \\ $$$$\Rightarrow\:{e}^{{i}\mathrm{4}{x}} −\mathrm{4}{ie}^{\mathrm{2}{ix}} −\mathrm{1}=\mathrm{0} \\ $$$${z}={e}^{{i}\mathrm{2}{x}} \:,\:{we}\:{have}\:{z}^{\mathrm{2}} −\mathrm{4}{iz}−\mathrm{1}=\mathrm{0}.. \\ $$$$……….. \\ $$