Question

Determine a forma trigonométrica do número complexo dada em cada item abaixo: • z = 2i• z = -1 + i

Answer 1

Explicação passo-a-passo:

a) $\sf z=2i$

=> Módulo

$\sf \rho=\sqrt{0^2+2^2}$

$\sf \rho=\sqrt{0+4}$

$\sf \rho=\sqrt{4}$

$\sf \rho=2$

Temos:

• $\sf sen~\theta=\dfrac{2}{2}~\Rightarrow~sen~\theta=1$

• $\sf cos~\theta=\dfrac{0}{2}~\Rightarrow~cos~\theta=0$

Assim, $\sf \theta=\dfrac{\pi}{2}~rad$

A forma trigomométrica é:

$\sf z=\rho\cdot(cos~\theta+i\cdot sen~\theta)$

$\sf z=2\cdot\Big[cos~\Big(\dfrac{\pi}{2}\Big)+i\cdot sen~\Big(\dfrac{\pi}{2}\Big)\Big]$

b) $\sf z=-1+i$

=> Módulo

$\sf \rho=\sqrt{(-1)^2+1^2}$

$\sf \rho=\sqrt{1+1}$

$\sf \rho=\sqrt{2}$

Temos:

• $\sf sen~\theta=\dfrac{1}{\sqrt{2}}~\Rightarrow~sen~\theta=\dfrac{\sqrt{2}}{2}$

• $\sf cos~\theta=\dfrac{-1}{\sqrt{2}}~\Rightarrow~cos~\theta=\dfrac{-\sqrt{2}}{2}$

Assim, $\sf \theta=\dfrac{3\pi}{4}~rad$

A forma trigomométrica é:

$\sf z=\rho\cdot(cos~\theta+i\cdot sen~\theta)$

$\sf z=\sqrt{2}\cdot\Big[cos~\Big(\dfrac{3\pi}{4}\Big)+i\cdot sen~\Big(\dfrac{3\pi}{4}\Big)\Big]$