### Teorija

Reizināšana un dalīšana
trigonometriskajā formā un eksponentformā

Reizinot divus kompleksus skaitļus ${z}_{1}={r}_{1}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\right)$ un ${z}_{2}={r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)$, reizinājuma $z={z}_{1}{z}_{2}$ modulis sanāk moduļu reizinājums un reizinājums sanāk argumentu summa:
$r={r}_{1}{r}_{2}$ un $\mathrm{\varphi }={\mathrm{\varphi }}_{1}+{\mathrm{\varphi }}_{2}$
Dalot divus kompleksus skaitļus ${z}_{1}={r}_{1}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\right)$ un ${z}_{2}={r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)$, dalījuma $z=\frac{{z}_{1}}{{z}_{2}}$ modulis sanāk moduļu dalījums un arguments sanāk argumentu starpība:
$r=\frac{{r}_{1}}{{r}_{2}}$ un $\mathrm{\varphi }={\mathrm{\varphi }}_{1}-{\mathrm{\varphi }}_{2}$

Pierādījums
Pieņemsim, ka doti divi kompleksie skaitļi trigonometriskajās formās: ${r}_{1}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\right)$ un ${r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)$
Vispirms aprēķināsim to reizinājumu, pēc tam dalījumu. (Tur izmantotas summas un starpības kosinusa un sinusa formulas.)
$\begin{array}{l}{z}_{1}{z}_{2}={r}_{1}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\right)\cdot {r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)=\\ ={r}_{1}{r}_{2}\cdot \left(\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}-\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)+i\left(\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\right)\right)=\\ ={r}_{1}{r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\left({\mathrm{\varphi }}_{1}+{\mathrm{\varphi }}_{2}\right)+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\left({\mathrm{\varphi }}_{1}+{\mathrm{\varphi }}_{2}\right)\right)\\ \\ \frac{{z}_{1}}{{z}_{2}}=\frac{{r}_{1}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\right)}{{r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)}=\\ =\frac{{r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)\cdot \left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}-i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)}{{r}_{2}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)\cdot \left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}-i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)}=\\ =\frac{{r}_{1}}{{r}_{2}}\cdot \frac{\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}+\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)+i\phantom{\rule{0.147em}{0ex}}\left(\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}-\mathit{cos}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}{\mathrm{\varphi }}_{2}\right)}{{\mathit{cos}}^{2}{\mathrm{\varphi }}_{2}+{\mathit{sin}}^{2}{\mathrm{\varphi }}_{2}}=\\ =\frac{{r}_{1}}{{r}_{2}}\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\left({\mathrm{\varphi }}_{1}-{\mathrm{\varphi }}_{2}\right)+i\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\left({\mathrm{\varphi }}_{1}-{\mathrm{\varphi }}_{2}\right)\right)\end{array}$

Svarīgi!
Pēc reizināšanas vai dalīšanas argumenta vērtība ir jāpiemēro nosacījumam $-\mathrm{\pi }<\mathrm{\varphi }\le \mathrm{\pi }$, pieskaitot $2\mathrm{\pi }k$, kur k ir vesels skaitlis.

Tādas pašas sakarības attiecas arī uz kompleksiem skaitļiem eksponentformā, kurus arī apraksta ar moduļa un argumenta vērtībām.
$\begin{array}{l}{r}_{1}{e}^{i{\mathrm{\varphi }}_{1}}\cdot {r}_{2}{e}^{i{\mathrm{\varphi }}_{2}}=\left({r}_{1}{r}_{2}\right){e}^{i\left({\mathrm{\varphi }}_{1}+{\mathrm{\varphi }}_{2}\right)}\\ \frac{{r}_{1}{e}^{i{\mathrm{\varphi }}_{1}}}{{r}_{2}{e}^{i{\mathrm{\varphi }}_{2}}}=\left(\frac{{r}_{1}}{{r}_{2}}\right){e}^{i\left({\mathrm{\varphi }}_{1}-{\mathrm{\varphi }}_{2}\right)}\end{array}$