Question Number 83287 by Power last updated on 29/Feb/20
Answered by mr W last updated on 29/Feb/20
$$\mathrm{sin}\:{x}={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−….. \\ $$$$\frac{\mathrm{sin}\:{x}}{{x}}=\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−….. \\ $$$$ \\ $$$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)=\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−…..\right)\right) \\ $$$$=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−…..\right)\right) \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} +{o}\left({x}^{\mathrm{4}} \right) \\ $$$$\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)=\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}−\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−…..\right)\right) \\ $$$$=−\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−…..\right)\right) \\ $$$$=−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} +{o}\left({x}^{\mathrm{4}} \right) \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)+\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)}{{ax}^{{n}} } \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} +{o}\left({x}^{\mathrm{4}} \right)}{{ax}^{{n}} } \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\pi^{\mathrm{2}} {x}^{\mathrm{4}} }{\mathrm{36}}+{o}\left({x}^{\mathrm{4}} \right)}{{ax}^{{n}} }\overset{!} {=}\mathrm{1} \\ $$$$\Rightarrow{n}=\mathrm{4} \\ $$$$\Rightarrow{a}=\frac{\pi^{\mathrm{2}} }{\mathrm{36}} \\ $$
Commented by Power last updated on 01/Mar/20
$$\mathrm{thanks} \\ $$