Question Number 14702 by tawa tawa last updated on 03/Jun/17
Answered by mrW1 last updated on 03/Jun/17
$$\left(\mathrm{sin}\:\mathrm{6}°+\:{i}\:\mathrm{cos}\:\mathrm{6}°\right)^{\mathrm{30}} \\ $$$$=\left(\mathrm{cos}\:\mathrm{84}°+\:{i}\:\mathrm{sin}\:\mathrm{94}°\right)^{\mathrm{30}} \\ $$$$=\mathrm{cos}\:\mathrm{30}×\mathrm{84}°+\:{i}\:\mathrm{sin}\:\mathrm{30}×\mathrm{84}°\: \\ $$$$=\mathrm{cos}\:\mathrm{2529}°+\:{i}\:\mathrm{sin}\:\mathrm{2520}°\: \\ $$$$=\mathrm{cos}\:\mathrm{7}×\mathrm{360}°+\:{i}\:\mathrm{sin}\:\mathrm{7}×\mathrm{360}°\: \\ $$$$=\mathrm{cos}\:\mathrm{0}°+\:{i}\:\mathrm{sin}\:\mathrm{0}°\: \\ $$$$=\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow{Answer}\:\left({a}\right) \\ $$
Commented by tawa tawa last updated on 03/Jun/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 03/Jun/17
$${de}'{muavr}\:{formula}: \\ $$$$\left({cosx}+{isinx}\right)^{\boldsymbol{{n}}} ={cos}\boldsymbol{{n}}{x}+{isin}\boldsymbol{{n}}{x} \\ $$$$\Rightarrow\frac{\left({isin}\mathrm{6}−{cos}\mathrm{6}\right)^{\mathrm{30}} }{{i}^{\mathrm{30}} }=−\mathrm{1}×\left({cos}\mathrm{6}−{isin}\mathrm{6}\right)^{\mathrm{30}} = \\ $$$$=−\mathrm{1}×\left({cos}\left(−\mathrm{6}\right)+{isin}\left(−\mathrm{6}\right)\right)^{\mathrm{30}} = \\ $$$$=−\mathrm{1}×\left({cos}\left(−\mathrm{180}\right)+{isin}\left(−\mathrm{180}\right)\right)= \\ $$$$=−\mathrm{1}×\left(−\mathrm{1}+\mathrm{0}\right)=+\mathrm{1}\:. \\ $$$${note}:\:\:{i}^{\mathrm{30}} =\left({i}^{\mathrm{2}} \right)^{\mathrm{15}} =\left(−\mathrm{1}\right)^{\mathrm{15}} =−\mathrm{1}\:.\blacksquare \\ $$
Commented by tawa tawa last updated on 04/Jun/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$