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if-xsina-cosa-n-q-x-2-1-r-find-r-




Question Number 73056 by mathmax by abdo last updated on 05/Nov/19
if (xsina+cosa)^n =q(x^2  +1)+r find r
$${if}\:\left({xsina}+{cosa}\right)^{{n}} ={q}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)+{r}\:{find}\:{r} \\ $$
Commented by mathmax by abdo last updated on 06/Nov/19
degr<2 ⇒r(x)=αx +β ⇒(cosa +xsina)^n  =q(x^2  +1)+αx+β  x=i ⇒(cosa+isina)^n  =αi +β ⇒cos(na)+isin(na)=αi +β  x=−i ⇒(cosa−isina)^n =−αi+β ⇒cos(na)−isin(a) =−αi +β ⇒  ⇒2cos(na) =2β ⇒β =cos(na) and 2isin(na)=2αi ⇒  α=sin(na) ⇒r(x)=sin(na)x +cos(na)
$${degr}<\mathrm{2}\:\Rightarrow{r}\left({x}\right)=\alpha{x}\:+\beta\:\Rightarrow\left({cosa}\:+{xsina}\right)^{{n}} \:={q}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)+\alpha{x}+\beta \\ $$$${x}={i}\:\Rightarrow\left({cosa}+{isina}\right)^{{n}} \:=\alpha{i}\:+\beta\:\Rightarrow{cos}\left({na}\right)+{isin}\left({na}\right)=\alpha{i}\:+\beta \\ $$$${x}=−{i}\:\Rightarrow\left({cosa}−{isina}\right)^{{n}} =−\alpha{i}+\beta\:\Rightarrow{cos}\left({na}\right)−{isin}\left({a}\right)\:=−\alpha{i}\:+\beta\:\Rightarrow \\ $$$$\Rightarrow\mathrm{2}{cos}\left({na}\right)\:=\mathrm{2}\beta\:\Rightarrow\beta\:={cos}\left({na}\right)\:{and}\:\mathrm{2}{isin}\left({na}\right)=\mathrm{2}\alpha{i}\:\Rightarrow \\ $$$$\alpha={sin}\left({na}\right)\:\Rightarrow{r}\left({x}\right)={sin}\left({na}\right){x}\:+{cos}\left({na}\right) \\ $$

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