Question

Sendo Z = 2√3 + 2i e W = –2 + 2√3i. o argumento do complexo resultado de Z/W é: a) 3π/2 b) 2π/3 c) 3π/4 d) 4π/3

Answer 1

Resposta:

$\text{letra C}$

Explicação passo-a-passo:

$z = 2\sqrt{3} + 2i\:\:\:\:\:\:\:w = -2 + 2\sqrt{3}i$

$\dfrac{z}{w} = \dfrac{2\sqrt{3} + 2i}{-2 + 2\sqrt{3}i}$

$\dfrac{2\sqrt{3} + 2i}{-2 + 2\sqrt{3}i} \times \dfrac{-2 - 2\sqrt{3}i}{-2 - 2\sqrt{3}i} = \dfrac{-4\sqrt{3} - 12i - 4i - 4\sqrt{3}i^2}{4 + 4\sqrt{3}i - 4\sqrt{3}i - 12i^2}$

$\dfrac{-4\sqrt{3} - 12i - 4i - 4\sqrt{3}(-1)}{4 + 4\sqrt{3}i - 4\sqrt{3}i - 12(-1)} = \dfrac{-16i}{16} = -i$

$\boxed{\boxed{\dfrac{z}{w} = -i}}$

$|\text{ z }| = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1$

$\text{cos }\Theta = \dfrac{\text{a}}{|\text{ z }|} = \dfrac{0}{1} = 0$

$\text{sen }\Theta = \dfrac{\text{b}}{|\text{ z }|} = \dfrac{-1}{1} = -1$

$\boxed{\boxed{\Theta = \dfrac{3\pi }{4}}}$