Ejemplos multiplicación números complejos en su forma polar o trigonométrica

Para multiplicar dos números complejos en su forma polar o trigonométrica se multiplican los módulos y se suman los argumentos.

Sean

$z_1=r_1\hspace{0.2cm}cis\hspace{0.2cm}\theta_1$ y $z_2=r_2\hspace{0.2cm}cis\hspace{0.2cm}\theta_2$ ,

entonces la multiplicación se define como:

$z_1z_2=r_1r_2\hspace{0.2cm}cis\hspace{0.2cm}(\theta_1+\theta_2)$

En los siguientes ejemplos efectuamos la multiplicación de los dos números complejos que se indican en su forma polar o trigonométrica:

Ejemplo 1:

$z_1=2\hspace{0.2cm}cis\hspace{0.2cm}35^\circ$ y $z_2=3\hspace{0.2cm}cis\hspace{0.2cm}10^\circ$

$z_1z_2=2\cdot3\hspace{0.2cm}cis\hspace{0.2cm}(35^\circ+10^\circ)$

$z_1z_2=6\hspace{0.2cm}cis\hspace{0.2cm}45^\circ$

Ejemplo 2:

$z_1=3\hspace{0.2cm}cis\hspace{0.2cm}20^\circ$ y $z_2=4\hspace{0.2cm}cis\hspace{0.2cm}110^\circ$

$z_1z_2=3\cdot4\hspace{0.2cm}cis\hspace{0.2cm}(20^\circ+110^\circ)$

$z_1z_2=12\hspace{0.2cm}cis\hspace{0.2cm}130^\circ$

Ejemplo 3:

$z_1=\sqrt{2}\hspace{0.2cm}cis\hspace{0.2cm}120^\circ$ y $z_2=\sqrt{2}\hspace{0.2cm}cis\hspace{0.2cm}40^\circ$

$z_1z_2=\sqrt{2}\cdot\sqrt{2}\hspace{0.2cm}cis\hspace{0.2cm}(120^\circ+40^\circ)$

$z_1z_2=2\hspace{0.2cm}cis\hspace{0.2cm}160^\circ$

Ejemplo 4:

$z_1=4\hspace{0.2cm}cis\hspace{0.2cm}150^\circ$ y $z_2=5\hspace{0.2cm}cis\hspace{0.2cm}60^\circ$

$z_1z_2=4\cdot5\hspace{0.2cm}cis\hspace{0.2cm}(150^\circ+60^\circ)$

$z_1z_2=20\hspace{0.2cm}cis\hspace{0.2cm}210^\circ$

Ejemplo 5:

$z_1=3\hspace{0.2cm}cis\hspace{0.2cm}260^\circ$ y $z_2=6\hspace{0.2cm}cis\hspace{0.2cm}30^\circ$

$z_1z_2=3\cdot6\hspace{0.2cm}cis\hspace{0.2cm}(260^\circ+30^\circ)$

$z_1z_2=18\hspace{0.2cm}cis\hspace{0.2cm}290^\circ$

Ejemplo 6:

$z_1=5\hspace{0.2cm}cis\hspace{0.2cm}300^\circ$ y $z_2=10\hspace{0.2cm}cis\hspace{0.2cm}30^\circ$

$z_1z_2=5\cdot10\hspace{0.2cm}cis\hspace{0.2cm}(300^\circ+30^\circ)$

$z_1z_2=50\hspace{0.2cm}cis\hspace{0.2cm}330^\circ$

Ejemplo 7:

$z_1=2\hspace{0.2cm}cis\hspace{0.2cm}45^\circ$ y $z_2=\pi\hspace{0.2cm}cis\hspace{0.2cm}100^\circ$

$z_1z_2=2\cdot\pi\hspace{0.2cm}cis\hspace{0.2cm}(45^\circ+100^\circ)$

$z_1z_2=2\pi\hspace{0.2cm}cis\hspace{0.2cm}145^\circ$

Ejemplo 8:

$z_1=\frac{1}{3}\hspace{0.2cm}cis\hspace{0.2cm}35^\circ$ y $z_2=3\hspace{0.2cm}cis\hspace{0.2cm}70^\circ$

$z_1z_2=\frac{1}{3}\cdot3\hspace{0.2cm}cis\hspace{0.2cm}(35^\circ+70^\circ)$

$z_1z_2=1\hspace{0.2cm}cis\hspace{0.2cm}105^\circ$

Ejemplo 9:

$z_1=\frac{1}{2}\hspace{0.2cm}cis\hspace{0.2cm}55^\circ$ y $z_2=3\hspace{0.2cm}cis\hspace{0.2cm}115^\circ$

$z_1z_2=\frac{1}{2}\cdot3\hspace{0.2cm}cis\hspace{0.2cm}(55^\circ+115^\circ)$

$z_1z_2=\frac{3}{2}\hspace{0.2cm}cis\hspace{0.2cm}170^\circ$

Ejemplo 10:

$z_1=\frac{1}{7}\hspace{0.2cm}cis\hspace{0.2cm}95^\circ$ y $z_2=49\hspace{0.2cm}cis\hspace{0.2cm}160^\circ$

$z_1z_2=\frac{1}{7}\cdot49\hspace{0.2cm}cis\hspace{0.2cm}(95^\circ+160^\circ)$

$z_1z_2=7\hspace{0.2cm}cis\hspace{0.2cm}255^\circ$