Question

Calcule o módulo e argumento dos seguintes complexos:
a) 2 + 2i
b) 1 – i
c) 3i
d) – 4 – 4i

Answer 1

Explicação passo-a-passo:

a) z = 2 + 2i

$|z|=\sqrt{2^2+2^2}$

$|z|=\sqrt{4+4}$

$|z|=\sqrt{8}$

$|z|=2\sqrt{2}$

$\text{sen}~\theta=\dfrac{2}{2\sqrt{2}}~\longrightarrow~\text{sen}~\theta=\dfrac{\sqrt{2}}{2}$

$\text{cos}~\theta=\dfrac{2}{2\sqrt{2}}~\longrightarrow~\text{cos}~\theta=\dfrac{\sqrt{2}}{2}$

$\text{arg}(z)=\dfrac{\pi}{4}~\text{rad}$

b) z = 1 - i

$|z|=\sqrt{1^2+(-1)^2}$

$|z|=\sqrt{1+1}$

$|z|=\sqrt{2}$

$\text{sen}~\theta=\dfrac{-1}{\sqrt{2}}~\longrightarrow~\text{sen}~\theta=\dfrac{-\sqrt{2}}{2}$

$\text{cos}~\theta=\dfrac{1}{\sqrt{2}}~\longrightarrow~\text{cos}~\theta=\dfrac{\sqrt{2}}{2}$

$\text{arg}(z)=\dfrac{7\pi}{4}~\text{rad}$

c) z = 3i

$|z|=\sqrt{0^2+3^2}$

$|z|=\sqrt{0+9}$

$|z|=\sqrt{9}$

$|z|=3$

$\text{sen}~\theta=\dfrac{3}{3}~\longrightarrow~\text{sen}~\theta=1$

$\text{cos}~\theta=\dfrac{0}{3}~\longrightarrow~\text{cos}~\theta=0$

$\text{arg}(z)=\dfrac{\pi}{2}~\text{rad}$

d) z = -4 - 4i

$|z|=\sqrt{(-4)^2+(-4)^2}$

$|z|=\sqrt{16+16}$

$|z|=\sqrt{32}$

$|z|=4\sqrt{2}$

$\text{sen}~\theta=\dfrac{-4}{4\sqrt{2}}~\longrightarrow~\text{sen}~\theta=\dfrac{-\sqrt{2}}{2}$

$\text{cos}~\theta=\dfrac{-4}{4\sqrt{2}}~\longrightarrow~\text{cos}~\theta=\dfrac{-\sqrt{2}}{2}$

$\text{arg}(z)=\dfrac{5\pi}{4}~\text{rad}$