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Eksāmens MATEMĀTIKĀ 12. KLASEI
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Teorija

Kāpināšana ar naturālu kāpinātāju un saknes

Ja kompleksais skaitlis z ir dots trigonometriskajā formā ($r\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\varphi }+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\varphi }\right)$) vai eksponentformā ($r{e}^{i\mathrm{\varphi }}$), tad kāpināšanai naturālā pakāpē n var iegūt šādas divas formulas (ja izmanto reizināšanas likumu):
$\begin{array}{l}{z}^{n}={r}^{n}\left(\mathit{cos}\left(n\mathrm{\varphi }\right)+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\left(n\mathrm{\varphi }\right)\right)\\ {z}^{n}={r}^{n}{e}^{\mathit{in}\mathrm{\varphi }}\end{array}$
Piemērs:
$\begin{array}{l}z=1+\sqrt{3}\phantom{\rule{0.147em}{0ex}}i\\ r=\sqrt{{1}^{2}+{\left(\sqrt{3}\right)}^{2}}=\sqrt{1+3}=2\\ \left\{\begin{array}{c}\frac{a}{r}=\frac{1}{2}=\mathit{cos}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\pi }}{3}\\ \frac{b}{r}=\frac{\sqrt{3}}{2}=\mathit{sin}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\pi }}{3}\end{array}\right\\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}⇒\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathrm{\varphi }=\frac{\mathrm{\pi }}{3}\\ z=2\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\pi }}{3}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\pi }}{3}\right)\\ \\ {z}^{7}={2}^{7}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\left(7\cdot \frac{\mathrm{\pi }}{3}\right)+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\left(7\cdot \frac{\mathrm{\pi }}{3}\right)\right)=\\ =128\phantom{\rule{0.147em}{0ex}}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\frac{7\mathrm{\pi }}{3}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\frac{7\mathrm{\pi }}{3}\right)=\\ =128\phantom{\rule{0.147em}{0ex}}\left(\frac{1}{2}+i\cdot \frac{\sqrt{3}}{2}\right)=\\ =64+64\sqrt{3}\phantom{\rule{0.147em}{0ex}}i\end{array}$
Svarīgi!
Vienādību ${\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\varphi }+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\varphi }\right)}^{n}=\mathit{cos}\phantom{\rule{0.147em}{0ex}}\left(n\mathrm{\varphi }\right)+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\left(n\mathrm{\varphi }\right)$ sauc par Muavra formulu.