Matematyka
mgr
2017-06-25 20:27:44
Proszę o pomoc w zadaniu;)
Odpowiedź
xXxGirlofdreamxXx
2017-06-25 23:25:28

[latex]z=1-i[/latex] [latex]|z|=sqrt{1+1}=sqrt{2}[/latex] [latex]egin{cases}cosvarphi=cfrac{1}{sqrt{2}}=cfrac{sqrt{2}}{2}\sinvarphi=-cfrac{sqrt{2}}{2}end{cases}Rightarrowvarphi=cfrac{7}{4}pi[/latex] [latex]z=sqrt{2}left(coscfrac{7}{4}pi+isincfrac{7}{4}pi ight)[/latex] [latex]z^{29}=(sqrt{2})^{29}left(cos50,75pi+isin50,75pi ight)=2^{14}sqrt{2}left(-cfrac{sqrt{2}}{2}+icfrac{sqrt{2}}{2} ight)=-2^{14}+i2^{14}[/latex] [latex](cosphi+isinphi)^{13}=cos13phi+isin13phi[/latex] [latex]|e^{iphi}|=|cosphi+isinphi|=sqrt{cos^2phi+sin^2phi}=sqrt{1}=1[/latex] [latex]z=1-3iRightarrowoverline{z}=1+3i[/latex] [latex]left|cfrac{z}{overline{z}} ight|=left|cfrac{1-3i}{1+3i} ight|=left|cfrac{(1-3i)(1-3i)}{10} ight|=left|-0,8-0,6i ight|=sqrt{0,8^2+0,6^2}=sqrt{1}=1[/latex] [latex]|z|=|1-3i|=sqrt{1+9}=sqrt{10}[/latex] [latex]|overline{z}|=|1+3i|=sqrt{1+9}=sqrt{10}[/latex]

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