Question Number 197052 by pete last updated on 07/Sep/23
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{sin}^{−\mathrm{1}} \sqrt{\mathrm{2}} \\ $$
Answered by Frix last updated on 07/Sep/23
$$\mathrm{sin}\:{x}\:=\sqrt{\mathrm{2}} \\ $$$${x}=\left(\mathrm{2}{n}−\frac{\mathrm{1}}{\mathrm{2}}\right)\pi\pm\mathrm{i}\:\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right) \\ $$$$\mathrm{You}\:\mathrm{get}\:\mathrm{this}\:\mathrm{using} \\ $$$$\mathrm{sin}\:{x}\:=\frac{\mathrm{e}^{\mathrm{i}{x}} −\mathrm{e}^{−\mathrm{i}{x}} }{\mathrm{2i}} \\ $$
Commented by pete last updated on 07/Sep/23
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$