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if-Z-1-1-i-and-Z-2-i-4-by-using-demover-find-Z-1-Z-2-




Question Number 113983 by mohammad17 last updated on 16/Sep/20
if Z_1 =1−i and Z_2 =i^4  by using demover find (Z_1 /Z_2 ) ?
$${if}\:{Z}_{\mathrm{1}} =\mathrm{1}−{i}\:{and}\:{Z}_{\mathrm{2}} ={i}^{\mathrm{4}} \:{by}\:{using}\:{demover}\:{find}\:\frac{{Z}_{\mathrm{1}} }{{Z}_{\mathrm{2}} }\:? \\ $$
Answered by Dwaipayan Shikari last updated on 16/Sep/20
(Z_1 /Z_2 )=((1−i)/i^4 )=1−i
$$\frac{{Z}_{\mathrm{1}} }{{Z}_{\mathrm{2}} }=\frac{\mathrm{1}−{i}}{{i}^{\mathrm{4}} }=\mathrm{1}−{i} \\ $$
Commented by mohammad17 last updated on 16/Sep/20
sir i want this by demover theorem
$${sir}\:{i}\:{want}\:{this}\:{by}\:{demover}\:{theorem}\: \\ $$
Answered by Dwaipayan Shikari last updated on 16/Sep/20
Z_1 =(√2)((1/( (√2)))−(1/( (√2)))i)=(√2)(cos(π/4)−isin(π/4))  Z_1 =i^4 =(cos(π/2)+isin(π/2))^4 =cos(4.(π/2))+isin(4.(π/2))=1  (Z_1 /Z_2 )=(((√2)(cos(π/4)−isin(π/4)))/1)=(√2)(cos(π/4)−isin(π/4))=1−i
$${Z}_{\mathrm{1}} =\sqrt{\mathrm{2}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{i}\right)=\sqrt{\mathrm{2}}\left({cos}\frac{\pi}{\mathrm{4}}−{isin}\frac{\pi}{\mathrm{4}}\right) \\ $$$${Z}_{\mathrm{1}} ={i}^{\mathrm{4}} =\left({cos}\frac{\pi}{\mathrm{2}}+{isin}\frac{\pi}{\mathrm{2}}\right)^{\mathrm{4}} ={cos}\left(\mathrm{4}.\frac{\pi}{\mathrm{2}}\right)+{isin}\left(\mathrm{4}.\frac{\pi}{\mathrm{2}}\right)=\mathrm{1} \\ $$$$\frac{{Z}_{\mathrm{1}} }{{Z}_{\mathrm{2}} }=\frac{\sqrt{\mathrm{2}}\left({cos}\frac{\pi}{\mathrm{4}}−{isin}\frac{\pi}{\mathrm{4}}\right)}{\mathrm{1}}=\sqrt{\mathrm{2}}\left({cos}\frac{\pi}{\mathrm{4}}−{isin}\frac{\pi}{\mathrm{4}}\right)=\mathrm{1}−{i} \\ $$
Commented by mohammad17 last updated on 16/Sep/20
thank you sir
$${thank}\:{you}\:{sir} \\ $$

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