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Question-14702




Question Number 14702 by tawa tawa last updated on 03/Jun/17
Answered by mrW1 last updated on 03/Jun/17
(sin 6°+ i cos 6°)^(30)   =(cos 84°+ i sin 94°)^(30)   =cos 30×84°+ i sin 30×84°   =cos 2529°+ i sin 2520°   =cos 7×360°+ i sin 7×360°   =cos 0°+ i sin 0°   =1    ⇒Answer (a)
$$\left(\mathrm{sin}\:\mathrm{6}°+\:{i}\:\mathrm{cos}\:\mathrm{6}°\right)^{\mathrm{30}} \\ $$$$=\left(\mathrm{cos}\:\mathrm{84}°+\:{i}\:\mathrm{sin}\:\mathrm{94}°\right)^{\mathrm{30}} \\ $$$$=\mathrm{cos}\:\mathrm{30}×\mathrm{84}°+\:{i}\:\mathrm{sin}\:\mathrm{30}×\mathrm{84}°\: \\ $$$$=\mathrm{cos}\:\mathrm{2529}°+\:{i}\:\mathrm{sin}\:\mathrm{2520}°\: \\ $$$$=\mathrm{cos}\:\mathrm{7}×\mathrm{360}°+\:{i}\:\mathrm{sin}\:\mathrm{7}×\mathrm{360}°\: \\ $$$$=\mathrm{cos}\:\mathrm{0}°+\:{i}\:\mathrm{sin}\:\mathrm{0}°\: \\ $$$$=\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow{Answer}\:\left({a}\right) \\ $$
Commented by tawa tawa last updated on 03/Jun/17
God bless you sir.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 03/Jun/17
de′muavr formula:  (cosx+isinx)^n =cosnx+isinnx  ⇒(((isin6−cos6)^(30) )/i^(30) )=−1×(cos6−isin6)^(30) =  =−1×(cos(−6)+isin(−6))^(30) =  =−1×(cos(−180)+isin(−180))=  =−1×(−1+0)=+1 .  note:  i^(30) =(i^2 )^(15) =(−1)^(15) =−1 .■
$${de}'{muavr}\:{formula}: \\ $$$$\left({cosx}+{isinx}\right)^{\boldsymbol{{n}}} ={cos}\boldsymbol{{n}}{x}+{isin}\boldsymbol{{n}}{x} \\ $$$$\Rightarrow\frac{\left({isin}\mathrm{6}−{cos}\mathrm{6}\right)^{\mathrm{30}} }{{i}^{\mathrm{30}} }=−\mathrm{1}×\left({cos}\mathrm{6}−{isin}\mathrm{6}\right)^{\mathrm{30}} = \\ $$$$=−\mathrm{1}×\left({cos}\left(−\mathrm{6}\right)+{isin}\left(−\mathrm{6}\right)\right)^{\mathrm{30}} = \\ $$$$=−\mathrm{1}×\left({cos}\left(−\mathrm{180}\right)+{isin}\left(−\mathrm{180}\right)\right)= \\ $$$$=−\mathrm{1}×\left(−\mathrm{1}+\mathrm{0}\right)=+\mathrm{1}\:. \\ $$$${note}:\:\:{i}^{\mathrm{30}} =\left({i}^{\mathrm{2}} \right)^{\mathrm{15}} =\left(−\mathrm{1}\right)^{\mathrm{15}} =−\mathrm{1}\:.\blacksquare \\ $$
Commented by tawa tawa last updated on 04/Jun/17
God bless you sir.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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