Question

Passe para a forma trigométrica de z=3i
Com resolução

Answer 1

$z=3i=0+3i\implies a=0,\,b=3.$

FORMA TRIGONOMÉTRICA (ou polar):

$z=\rho[\mathrm{cos}\theta+(\mathrm{sen}\theta)i];~\begin{cases}\rho=|z|=\sqrt{a^2+b^2}\\\mathrm{cos}\theta=\frac{a}{\rho}\\\mathrm{sen}\theta=\frac{b}{\rho}.\end{cases}$

$\therefore$

$\rho=\sqrt{0^2+3^2}=\sqrt{0+9}=\sqrt{9}=3.$

$\mathrm{cos}\theta=\dfrac{0}{3}=0.$

$\mathrm{sen}\theta=\dfrac{3}{3}=1.$

Cosseno e seno de $\theta$ que resultam em, respectivamente, 0 e 1, só pode ser $\theta = 90^\circ=\frac{\pi}{2}.$

$\therefore$

$z=\rho[\mathrm{cos}\theta+(\mathrm{sen}\theta)i]\implies$

$\implies \underline{\boxed{z=3\bigg[\mathrm{cos}\frac{\pi}{2}+(\mathrm{sen}\frac{\pi}{2})i\bigg]}}\,.~\leftarrow~\text{forma trigonometrica.}$