Question Number 210456 by peter frank last updated on 09/Aug/24
Commented by mr W last updated on 09/Aug/24
$${who}\:{knows}\:{what}\:{is}\:\mathrm{2}\:{and}\:{what}\:{is}\:{z} \\ $$$${when}\:{you}\:{write}\:{them}\:{identically}? \\ $$
Commented by Spillover last updated on 09/Aug/24
Commented by peter frank last updated on 10/Aug/24
$$\mathrm{thanks}\:\mathrm{for}\:\mathrm{correcting}\:\mathrm{the}\:\mathrm{question} \\ $$
Commented by lepuissantcedricjunior last updated on 10/Aug/24
$$\boldsymbol{{z}}=\boldsymbol{{cos}\theta}+\boldsymbol{{isin}\theta}=\boldsymbol{{e}}^{\boldsymbol{{i}\theta}} \\ $$$$\boldsymbol{{mtq}}\:\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{z}}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}โ\boldsymbol{{tan}}\left(\boldsymbol{\theta}/\mathrm{2}\right)\right) \\ $$$$\boldsymbol{{on}}\:\boldsymbol{{a}}\:\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{z}}}=\frac{\mathrm{1}}{\boldsymbol{{e}}^{\boldsymbol{{i}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}โ\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)} +\boldsymbol{{e}}^{\boldsymbol{{i}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}+\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\boldsymbol{{e}}^{โ\boldsymbol{{i}}\frac{\boldsymbol{\theta}}{\mathrm{2}}} }{\mathrm{2}\boldsymbol{{cos}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)\:}=\frac{\boldsymbol{{cos}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)โ\boldsymbol{{isin}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)}{\mathrm{2}\boldsymbol{{cos}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)} \\ $$$$\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}โ\boldsymbol{{itan}}\left(\frac{\boldsymbol{\theta}}{\mathrm{2}}\right)\right) \\ $$$$…………..{prof}\:{cedric}\:{junior}…………… \\ $$
Commented by Spillover last updated on 10/Aug/24
$${good} \\ $$
Answered by Spillover last updated on 09/Aug/24
Commented by peter frank last updated on 10/Aug/24
$$\mathrm{thanks}\:\mathrm{spillover} \\ $$
Answered by mm1342 last updated on 10/Aug/24
$$\frac{\mathrm{1}}{\mathrm{1}+{z}}=\frac{\mathrm{1}+{cos}\thetaโ{isin}\theta}{\left(\mathrm{1}+{cos}\theta\right)^{\mathrm{2}} +{sin}^{\mathrm{2}} \theta}=\frac{\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)โ\mathrm{2}{isin}\left(\frac{\theta}{\mathrm{2}}\right){cos}\left(\frac{\theta}{\mathrm{2}}\right)}{\mathrm{4}{cos}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}โ{itan}\left(\frac{\theta}{\mathrm{2}}\right)\right)\:\checkmark \\ $$$$ \\ $$
Commented by peter frank last updated on 10/Aug/24
$$\mathrm{thanks}.\mathrm{mm1342} \\ $$
Commented by Spillover last updated on 10/Aug/24
$${good} \\ $$