Question

Calcule o produto de [0.7.(cosπ\2+ i.senπ\2).[2,6.(cosπ\3+ i.senπ\3)]

Answer 1

Produto de números complexos na forma trigonométrica

Sendo $z_{1}=\rho_{1}.[\cos(\theta_{1}) +i\sin(\theta_{1})]$ e$z_{2}=\rho_{2}.[\cos(\theta_{2}) +i\sin(\theta_{2})]$ então $z_{1}.z_{2}=\rho_{1}.\rho_{2}[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})]$

Dados

$\rho_{1}=0,7$

$\rho_{2}=2,6$

$\theta_{1}=\frac{\pi}{2}$

$\theta_{2}=\frac{\pi}{3}$

cálculos auxiliares:

mmc(3,2)=6

$\dfrac{\pi}{2}+\dfrac{\pi} {3}=\dfrac{3\pi+2\pi}{6}=\dfrac{5\pi}{6}$

Substituindo temos

$z_{1}.z_{2}=0,7.2,6[\cos(\frac{\pi}{2}+\frac{\pi}{3})+i\sin(\frac{\pi}{2}+\frac{\pi}{3})]$

$\boxed{\boxed{\mathsf{z_{1}.z_{2}=1,82[\cos(\frac{5\pi}{6})+i\sin(\frac{5\pi}{6})]}}}$