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Forma trigonometrică a unui număr complex | Lectii-Virtuale.ro - Sa Se Scrie Sub Forma Trigonometrica Numerele
Forma trigonometrică a numărului complex z este: ... unde ... Numerele r și t se numesc coordonate polare (r este raza polară, iar t argumentul polar). Mulțimea ...
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\,\,4\left(%20{\cos%20\left(%20{\frac{\pi%20}{2}}%20\right)%20+%20i\sin%20\left(%20{\frac{\pi%20}{2}}%20\right)}%20\right)%20\\%20%20Consideram\,\,numarul\,\,cautat\,z\,%20=%20a%20+%20bi\,%20\\%20%20Forma\,\,trigonometrica\,\,a\,\,numarului\,\,z\,\,este:%20\\%20%20z%20=%20r\left(%20{\cos%20\lambda%20%20+%20i\sin%20\lambda%20}%20\right)\,\,unde:%20\\%20%20r%20=%20|z|%20\\%20%20Le\,\,pun\,\,unele\,\,sub\,\,altele\,pentru\,\,a\,\,vedea\,\,asemanarile:%20\\%20%20\left.%20\begin{array}{l}%20z%20=%20r\left(%20{\cos%20\lambda%20%20+%20i\sin%20\lambda%20}%20\right)%20\\%20%20z%20=%20\,4\left(%20{\cos%20\left(%20{\frac{\pi%20}{2}}%20\right)%20+%20i\sin%20\left(%20{\frac{\pi%20}{2}}%20\right)}%20\right)%20\\%20%20\end{array}%20\right)%20\Rightarrow%20\left\{%20\begin{array}{l}%20r%20=%204%20=%20\sqrt%20{a^2%20%20+%20b^2%20}%20%20\Rightarrow%20a^2%20%20+%20b^2%20%20=%2016%20\\%20%20\lambda%20%20=%20\frac{\pi%20}{2}%20\\%20%20\end{array}%20\right.%20\\%20%20Pe\,\,de\,\,alta\,parte:%20\\%20%20\left\{%20\begin{array}{l}%20\sin%20\lambda%20%20=%20\frac{b}{r}%20\\%20%20\cos%20\lambda%20%20=%20\frac{a}{r}%20\\%20%20\end{array}%20\right.%20\Rightarrow%20\left\{%20\begin{array}{l}%20b%20=%20r\sin%20\lambda%20%20\\%20%20a%20=%20r\cos%20\lambda%20%20\\%20%20\end{array}%20\right.%20\Rightarrow%20\left\{%20\begin{array}{l}%20b%20=%20r\sin%20\frac{\pi%20}{2}%20\\%20%20a%20=%20r\cos%20\frac{\pi%20}{2}%20\\%20%20\end{array}%20\right.%20\Rightarrow%20\left\{%20\begin{array}{l}%20b%20=%20r%20\\%20%20a%20=%200%20\\%20%20\end{array}%20\right.%20\Rightarrow%20z%20=%20ri%20\Rightarrow%20z%20=%204i%20\\%20%20\end{array}\])
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